You have to bring those bits and pieces together, put them into the right shapes and fill in the gaps to get a model that actually performs. This is what post-training is all about. It's not at all a trivial thing.
Reasoning, tool use, agentic behavior - all of those are post-training performance gains. Getting a good well trained base model is putting your foot in the door of frontier performance - post-training is how you actually get inside.
See: GPT-4.5 vs o1. One went for "build a bigger better more capable base model", the other went for "take the old base and post-train it for advanced capabilities". The results: a wider base with basic post-training loses to a narrower base with advanced post-training. Or, hell: GPT-3 vs GPT-3.5. One was largely a research lab curio, and the other kicked off the AI revolution as we know it.
The gains compound. Getting a better base model with the same type of post-training helps, see: the jump from Opus to Mythos/Fable. But post-training techniques account for a lot of the performance juice.
And yes, reasoning trace post-training distillation is "genuine distillation". As is logit distillation in pre-training. "Distillation" isn't a single training recipe that you have to follow to a tee - it's a large group of training methods. I've seen plenty of wacky things like inverse distillation bootstrap and post-training self-distillation that use distillation in strange ways at different stages of the training run to get results.