This sounds like a classic case of imposter syndrome, which affects many mathematicians. Consider the OP's first paragraph.
I am an early career researcher working in an area of "hard" analysis but this is a fairly broad question. My technical skills are likely below par and my greatest hindrance to my research output. I often see authors in a similar point in their careers in my subfield that write on a higher technical level. I see computations, elementary technical lemmas and estimates that I would be unable to come up with and carry out if I were pursuing the project myself or if I were simply already told the statements and asked to prove them myself.
A sizeable proportion of mathematicians have had feelings just like you describe above. Most mathematicians were among the strongest students in primary school, but due to the narrowing funnel, find themselves surrounded by strong peers in graduate school. Because we don't see others' struggles, when we see someone do something that we would struggle with, it tends to look easy for them. But in fact, there are many things you could do, that would look very hard to someone else (even another student in your program). Managing your mental well-being is critical for success as a mathematician, so it's wise to read up on imposter syndrome, recognize that these feelings are normal and are detrimental to success, and take concrete steps to combat these feelings, like spending time with friends away from work, taking on hobbies where you can see your own skills (especially new hobbies, where you can rapidly increase from novice level to an intermediate level), recognize when negative emotions are rising (and deflect them mindfully), and give yourself affirmations. Here are some great threads about imposter syndrome from academia.SE:
https://academia.stackexchange.com/questions/11765/how-to-effectively-deal-with-imposter-syndrome-and-feelings-of-inadequacy-ive
https://academia.stackexchange.com/questions/41067/how-do-i-know-if-i-am-truly-prepared-for-graduate-school-in-mathematics
https://academia.stackexchange.com/questions/187561/math-postdoc-position-for-a-very-mediocre-new-phd-graduate/187576#187576
https://academia.stackexchange.com/questions/201849/new-faculty-already-lagging-on-responsibilities-bad-any-advice-on-coming-back
https://academia.stackexchange.com/questions/81005/how-do-i-reassure-myself-that-i-am-a-worthy-candidate-for-a-tenure-track-positio
Ok let's move on to the value of technical work in math.
I am aware that longer and more technical mathematics is not necessarily better mathematics, but I think it's undeniable that having strong technical ability is a boon. I know that the big picture is important but it's somewhat frustrating when attending a talk or discussing a paper and parts of papers are just dismissed as "technical ends". After all, they're a necessary part to doing research and strengthening this skill helps build mathematical confidence and make the process a touch more enjoyable.
I do research in a very technical field. After PhD training, most research professors could fill out the technical steps. Thus, in a seminar or conference talk, it's best to focus on what it was all good for, i.e., the new results, and not the technical steps it took to get there. However, all the technical steps are in the paper. As an early career researcher, I used to focus a lot on the technical steps when I presented the work, because that's where I'd put most of my time. It took a while to figure out which bits were "standard" (because I kept doing the same kind of technical thing over and over again in different papers, and realized my audience would have had similar experiences) and which bits were subtle and deserving of time in a seminar talk. At the beginning, it's all hard. Over time, through repetition and positive feedback (from colleagues, seminar audiences, referees, and the publication process) many aspects get easier and then each new paper can have one or two new ideas, unlike the first papers where it felt like everything was new. Keep at it! It might help to keep a document for yourself about which technical things seem to come up over and over, and which types of arguments you find yourself getting better at (this is a kind of affirmation).
I also recognize that the way I've referred to technical skill might make it sound like a one size fits all tool that immediately knocks out any problem. But the notion of general technical skill feels different from that of having collected a broad toolkit. Perhaps the essence of this question is about cleverness and problem solving.
A broad toolkit comes through years of practice and experience. You should not feel like it's a failure not to have one right away. Plenty of excellent mathematicians had a fairly limited bag of tricks, and it was more than enough to publish lots of papers. Focus on doing one thing really well, and then you can expand your repertoire one paper at a time for the rest of your career.
(1) For those who do very technical work, how does one lay a good foundation for building technical skills in ones research area? It does not feel like graduate coursework and prelim exams/orals provided this, nor am I sure if they're supposed to. I presume for most this base is built in graduate school and undergraduate study. Is this done by going through a foundational graduate text in the field and working through all the examples, propositions, theorems, and exercises 10 times over? Is this done by applying a similar process to research papers?
For most subfields of math, the graduate coursework will not be enough to do novel and interesting research. If it were, then thousands of mathematicians per year would be able to do what you do. That's why, after graduate coursework, you start reading research papers, mastering their techniques, and tunneling towards the edge of what is known, so that you can push beyond that threshold in your PhD thesis. You also should not feel like you need to work though an entire book ten times over. The book (really, set of books, because it's rare that there's one unique treatment that contains everything you might ever need) is the foundation. Its purpose is to give you enough skill to read the papers. If something in the paper refers to something in the book that you missed or forgot, then that triggers you to refresh yourself on that one portion of the book. Because of the "tunneling" mentioned above, it's possible that you'll eventually forget a lot of things you learned in your graduate qualifying exam courses, because they are not directly relevant to your future research. You'll probably remember most, but not all, of the things in the qual course in your specific area. For example, I'm a homotopy theorist, and studied Allen Hatcher's book to pass my graduate quals. Chapter 4 is about homotopy theory. I'm pretty sure there are a few very specific things in there, that seem niche when compared to my active line of research, that I've forgotten. And that's totally fine. If I ever needed them, I'd know where to look and I'd learn it a lot faster the second (or third or fourth) time around.
(2) How does one refine and improve their technical skills throughout their career? Is this skill built by brute forcing your way through new difficulties that arise and doing a million wrong computations before you find a good one? Do you teach to help reinforce fundamentals? Do you understand your technical lemmas well but have difficulty with those of other authors? I've always wondered how mathematicians with strong technical skills have reached their level.
Yes to all the above. At the beginning it can feel like brute force. Over time, you recognize the patterns (this is a huge part of math, and your advisor can help). Later computations are easier because they reduce to earlier ones, or similar techniques that you've seen before. Everything gets easier as the words, concepts, and techniques become more familiar and you are able to focus your full attention on just what's new in the current situation compared to all the others you've dealt with before. Teaching certainly helps reinforce fundamentals. So does MathOverflow and that's one reason a lot of folks answer questions on here (as well as helping junior people and "paying forward" the help we got when we were junior). Most people at all career stages have trouble understanding others' technical lemmas. The only way to truly understand something is to (re)invent it yourself. Examples help a ton, to see how someone else's technical lemma unfolds and what it does conceptually. Lastly, to reiterate, with each problem you solve and each paper you write, you get stronger, both with the techniques that you use over and over again, and with new techniques needed to solve the new problems. Explaining your work to others also forces you to understand it more deeply. Try not to pay attention to others making it look easy. Believe me, it's not easy for them. More importantly, even if it was easy for them, that should have nothing to do with your experience. We do math because we crave understanding. So recognize that it'll be hard, and just work through one step at a time, one concept at a time, and one problem at a time, and know that it'll get easier.
