Ah, beans. Was hoping for something more applicable to desktop processors. Neat idea though.
Edit: I missed the callout about laptops and tablets as I was skimming.
In my research I have found that the thermal paste is, paradoxically, the least thermally conducting part of a PC heat pipeline. So, use as little of it as possible, to avoid adding thickness.
Or use one made of a metal alloy:
Well, the second least conducting after what is meant to replace: Air.
And it makes sense, if it were more thermally conducting, then you should just make the entire cooler out of thermal paste.
Also, using too much thermal paste is not an issue. What matters is mounting pressure and geometry. Replacing the bendy Independent Loading Mechanism (ILM) with an aftermarket square bracket. If you want to go further you can lap the IHS or even delid for direct-die cooling.
Delid for liquid metal TIM in between the IHS and die made sense for a few years, but Intel has switched backed to solder, so the only gain to be had with delidding these days is with direct die.
You might be interested in watching this: <https://www.youtube.com/watch?v=EUWVVTY63hc> and maybe also this: <https://www.youtube.com/watch?v=nWu2tcm4wL8>. Do note that the videos contain graphs which summarize the findings.
Permanently attaching a cooler to the CPU itself doesn't really seem like a viable option to me. Mounting it to the motherboard would be a nightmare, and you'd end up creating a one-size-fits-nobody cooling solution. Nobody would be happy with the cooler you end up choosing.
In reality, cooling paste really isn't that big of a deal.
Don't do this. Far far worse than having too much thermal paste is having too little. Assuming you are using non-conductive paste(which most these days are), be liberal with the thermal paste. As long as you stay within reason it pretty much doesn't matter thermally how excessive you apply, especially when applying on a heat spreader.