Peltier cooling has been tried and found to be a non-starter. The problem is that they only move the thing that needs to be cooled a few millimetres; one is still left with the task of actually getting rid of the heat, which falls back to traditional heatsinks + fans. I saw people experimenting with such things - along with actual, honest-to-goodness evaporative coolers, i.e. miniature refrigerators - back in the early days of YouTube. The reason we don't see them now is that heatpipes were so much better.
Peltier devices are frankly terrible in terms of how much heat they can move for the amount of power they need. You can compare this with using mechanical devices (i.e. steam turbines and other heat engines) to extract power from a heat differential vs. using Seebeck devices. With our current technology they just work so much better than solid-state devices. The difference is significant enough for people (NASA I think) to seriously consider using Stirling engines, which have moving parts, over thermoelectric devices on space missions, despite the fact that one can't fix the former if it breaks down and the latter probably never would.
Just an aside... back in the really early days no-one needed to care about hot chips in laptops because they were basically tower cases with a handle attached. A little later on we were still using chips that gave off very little heat (386's didn't need any cooling at all, didn't they? I forget.)
When processing power started to really take off... again, no-one really cared because AT/ATX cases were roomy enough for air cooling and no-one expected desktop performance from a laptop** anyway.
Now we find ourselves in a situation where we're trying to cram a relatively modest thermal load into an increasingly small space. We're still using forced-air cooling because there's really nothing else better that can fit in the space we have.
**It's been said that the word "laptop" fell out of favour among marketing types at Apple because the metal cases often made them too hot to comfortably use as such. My PowerBook G4 could be very hot in the middle where the RAM slots were so I think there's a grain of truth there.