Thanks to both Pavlov and White there is now an almost complete list of "critical points" in the history of homotopy theory.

There are a few items that perhaps should make it into the list, for instance  the Archaic Period: Betti introduced Betti's numbers  [here][1] in 1870 -incidentally, as simple as they are, Betti's numbers continue to play a dramatic role in applied math, for instance in Persistent Homology. 

 I would suggest one tiny emendation: extend your 50 years slightly, by five years, to include a true **Annus Mirabilis** (1967), namely the work of Quillen on **Model Categories**.




 Let me say why I feel this is indeed a breakthrough and the beginning of modern era (some of the work after 2005 is properly speaking not modern, but post-modern, see below). 

Before Quillen, in  1958 Steenrod and Eilenberg managed to get the **Grand Unification of Cohomology**, certainly a major breakthrough in the series of "foundational efforts" in Algebraic Topology. 


In a way Quillen tried to do the same for homotopical algebra, by introducing a sets of axioms for "doing homotopy " in a category.  The key notion here is **weak equivalence**, ie a sets of maps in the ambient cat which   contain all the isomorphisms.

 This simple step is a foundational paradigm change, because it tells us WHAT Homotopy is all about: 


**we move from equality (set theory)_ to isomorphism (category theory) to equivalence (homotopy theory**). 






Quillen add some axioms on  formal fibrations, cofibrations, to compute the so-called homotopy limits and colimits, ie lims and colims "up to homotopy" 




NOTE before quillen folks knew about hom lim and hom colims: start with a cat with a model structure, and "localize" it, ie formally invert all the weak equivalences. The new cat, called the homotopy category, is is general not well behaved as far as standard lims and colims: one has thus to introduce a new kind of universal objects appropriate for the homotopic context. 

So, from the Annus Mirabilis begins a new chapter, but it does not end there. As a result of Quillen's  shift, now many many "things" that were not under the rubric of  homotopy theory, structures that are not even topological, acquire an  homotopic flavor. 


Funny enough, one of these is category theory itself: Cat, the category of small cats, has a default model structure, where weak equivalences are simply cats equivalence. 

There were several attempts to generalize and expand on the look at Homotopy given by model structures, but if we focus on true radically new insights, here it goes:

back in the golden era, a standard homotopy was simply a continuous deformation, so essentially an invertible path  between maps. In Quillen you start with the weak equivalences, but let us go back to the continuous deformation: if the cat where I want to introduce my weak equivalences happens to be a 2-cat, and I look at the groupoid of 2-maps therein, I have my continuous deformations. 



So, here is the key insight that migrates from the modern approach to the post-modern: do not look at a single cat alone, but see it as as only the ground floor of a higher and higher groupoid (paths, paths of paths, etc.) Rather than  being the bedrock of Homotopy, model structures become models, or presentations, of the REAL OBJECT of Homotopy, the **invariant infinity groupoid** in all its splendor. That basic insight is already in Grothendieck, around 1983, maybe earlier, but has blossomed into an entire field thanks to Voevodsky, Lurie, Resk, etc.

What is fascinating, is that the post-modern era is not simply foundational, but admits also a foundationalist approach: a large swath of  mathematics in principle can be seen from this angle, of "getting rid of equality", and replacing it with equivalences and their  higher versions. . 

[1]: https://link.springer.com/article/10.1007/BF02420029
  [2]: https://en.wikipedia.org/wiki/Rational_homotopy_theory