It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression is that this field is not yet completely settled, and so one might expect there to have been significant simplifications of this material since.
I know of one possible example (though I am not sure if it is best described as a simplification, rather than an alternative route), namely the proof of the equivalence between quasicategories and simplicial categories given in https://arxiv.org/abs/0911.0469. What other foundational advances have happened after the publication of HTT?
In Higher Topos Theory, Lurie accomplishes many things. Let me highlight a few:
A study of the Joyal model structure and comparison to the Bergner model structure.
A study of cartesian fibrations and straightening / unstraightening, the $\infty$-categorical analog of the Grothendieck construction. This is often viewed as the technical heart of Lurie's theory, since cartesian fibrations are used systematically to avoid writing down all the higher coherence data involved in $qCat$-valued functors.
A development of the fundamental notions of category theory -- (co)limits, Kan extensions, cofinality, etc, allowing one to "do category theory" in the $\infty$-categorical setting.
A development of the theory of presentable $\infty$-categories. The point here is to get access to (the most important instances of) Freyd's adjoint functor theorem in the $\infty$-categorical setting, and in particular the theory of localizations.
The theory of (Grothendieck) $\infty$-toposes.
In the context of foundations, maybe it's worth also mentioning some of the contents of Higher Algebra:
The Barr-Beck monadicity theorem. I tend to think of this, along with the adjoint functor theorem as "the only real theorems" of basic ordinary category theory.
A theory of operads, allowing one to "do algebra" $\infty$-categorically.
The theory of stable $\infty$-categories, playing roughly the roles of abelian categories and triangulated categories in the $\infty$-categorical setting.
So where are we today in comparison? Well, this is going to be woefully inadequate, but let me mention a few things:
Emily Riehl and Dominic Verity rework the theory of cartesian fibrations, (co)limits / Kan extensions, and monadicity, in a model-independent framework making several advances:
Their theory allows one to do all these things not just in the quasicategory model, but at the same time in, e.g. complete Segal spaces, along with several other models. Model comparison results show that corresponding notions in different models agree.
Not only does one obtain the same same statements in these other models, but the proofs are now the same across all these different models.
Not only are the proofs the same, but the definitions, theorems, and proofs for much of the theory are cast as natural adaptations of the corresponding notions in 1-category theory.
There are pedagogical advantages -- if you know ordinary category theory, this makes $\infty$-category theory much more accessible today. There are technical advantages -- you can use results in many more models than just quasicategories. And there are philosophical advantages -- it's much more clear that the theory of $\infty$-categories is in some sense (1) unique, rather than model-dependent and (2) inevitable as a generalization of ordinary category theory. Their work is spread across a number of papers, but they are writing a book consolidating their approach.
Nguyen, Raptis, and Schrade have proven the general adjoint functor theorem in the $\infty$-cateogorical context, which in principle subsumes much of the theory of presentable $\infty$-categories. But the theory of presentable $\infty$-categories is still essential in its own right and I'm not sure to what extent one can simplify the theory with this result.
There have been various model comparison results, e.g. here.
Lots more. I hope others will add answers adding more of these / expanding on this glib treatment!
The study of $\infty$-categories is still in its infancy, and likely there are are many fundamental facts about them which will one day appear to be of foundational significance which we have at most glimpsed hints of today. I have in mind various facts about Goodwillie calculus...
A significant technical improvement has been found by J. Shah in the theory of Kan extensions. Unfortunately I do not know of an exposition that does only the classical case, but reading the proof of theorem 10.3 in