I can only speak of my experience. I was first taught convergence of sequences and then later on, the definition of limits with $\varepsilon-\delta$ and its equivalent form involving sequences. I still find this most intuitive, but then this opinion is clearly heavily influenced by my upbringing.
For disclosure, I received my math education behind the Iron Curtain, and this model was employed by most (now former) communist countries. During that time curricula in most of those countries were devised by influential mathematicians who could also communicate math very well. (For example, in USSR Kolmogorov was deeply involved in shaping math education. He even wrote some high-school textbooks that were widely used.)
What I am attempting to communicate here is that the sequences-first system was adopted by informed mathematicians who cared about math education, it was tested on a large scale (tens if not hundreds of millions of students) for a long time (several decades). Arguably this system has produced good results.
Terry Tao's textbook on analysis (which I like very much for several reasons) also relies on a sequences-first approach.