Yes, you're correct. I think the evidences of evolution in natural history make the idea of evolution a fact.
However, my conclusion is drawn by inference from what little I know about the whole of Natural History, coming from what I've read, studied or critically thought about the topic of evolution.
So, what I have in my mind as an explanation is not a step by step, blow by blow transcription of "A Story" about whichever species of fish supposedly began the long haul in slowly providing the starting point of lineages leading to later species' development. and conveniently eventuating to you and me today, sitting as we are, typing with our evolved digits and sending our musings into the E-World.
As for Punctuated Equilibrium, I'm going to go by what both Stephen Jay Gould and Niles Eldridge have said about their own perspective on the theory. I can't say that it's contradictory to the Theory of Evolution since......well....they have both been secular Evolutionists and simply offer a modified explanation of the pacing of evolution. And if there are punctuated points in the pacing and direction of the natural history of evolution, as opposed to there be a clearly defined gradualism in the overall evolutionary record, then yeah.........I'm going to decline from offering a contrived, morphological 'tale' as a descriptive explanation. Evolution isn't a fact due to there be a clear account of the connection of details "proving" the evolutionary connections. No, it's a fact because, historically speaking, the fragments and diverse forms of scientific evidence are so numerous that we are firmly nudged to make inferences to this conclusion in a rational, justified way.
Of course, God could have directed it all all along, but even if He did, that doesn't mean we'd necessarily see clearly identifiable evidence of design in the geological, paleontological, archaeological, biological and genetic records, among others that are available through the sciences.
I mean, here's the upshot in this: if you personally would rather believe in a more Lamarckian and/or Theologically Progressive and directed form of evolution over the long-haul of millions upon millions of years, then be my guest. I won't knock you for it.