It was to beat the globalists to the punch.
The globohomo plan was to use the vaccine along with its accompanying passport industry to cement its grip of authority across the world. According to their plan, the vaccine would have taken years to develop, after which people overly weary of the lockdowns would have gladly accepted it along with any conditions attached.
By developing the vaccine so soon, this completely threw the plan out of whack. The passport industry, although ramping up, wasn't ready yet. They tried delaying the end of lockdowns long as they could, but people, after receiving vaccinations, were growing restless, and they couldn't avoid easing restrictions despite the passport system not being ready. Finally, after several months after lockdowns ended, they went ahead with passports anyway in certain governments (like NY) wherever the local despot was crazy enough to try it.
But it's a waste. People will now not accept passports after already opening up, the justification is too weak. It will only be a matter of time before the feeble attempts fail.
THIS was the true purpose of Warp Speed -- to beat the passport system to the punch, thereby nullifying it. Trump playing up the vaccine is simply to establish that that was all that was needed -- no passport necessary. The passport was always the lynchpin needed for globalists to cement control
You're correct in that Trump wouldn't have been able to make them do anything they weren't willing to (or prevent them from doing what they wanted). But he was able to convince the right players -- vaccine companies -- to take the bait (of pure greed) in order to throw off their plan just enough so that it doesn't work out.