Sounds like it's used regularly by some of the people here. Maybe less risk when using narrow bits?
Unlike blunt needles and wire, drill bits are designed to cut. If sized properly (same size as the hole, so will barely clear it) any misalignment, and you'll end up removing some or all of the copper from the hole itself. Possibly tearing a pad off as it exits. Not the best odds, so not good practice generally speaking.
Even assuming partial melting, if it was possible to pull the cap out won't it be similarly possible to later mechanically push the solder out?
Possible? Yes.
It depends on the temp and force applied after inserting some tool (tool can conduct heat away). So you could insert the tool, say a piece of wire or component leg, and it solidify rather than clear the hole out (you're boned without more heat if this happens). Even as it enters the plastic range, use too much force and you'll damage the board (lifted or torn traces). It's a balancing act to pull off, as you're flirting with the absolute minimal heat requirement for the given conditions, and there's considerable risk involved.
Simple solution is more heat.
Hair dryer, electric skillet, heat gun, or oven can help you with this, and you don't need to get anywhere near melting point. As mentioned before, 100C will do wonders. Just work quickly so it doesn't cool off too much if you can't apply preheating while soldering or desoldering.
But this bring me to what AG6QR said...
the remaining ones has come clean using a little piece of resistor lead, and heating the hole and lead with an iron while pushing the lead through the hole.
Do you specifically mean that component leads are better heat conductors than needles? I guess the end is wider, at the very least.
FWIW, even the best irons need a preheater if the PCB has enough layers sucking off the heat.
If it works well enough for some of the people here, on similar boards, using similar basic gear, I do wonder what's different.
I think you're missing a critical point, and that's that the solder is fully molten when using wire, component leads, needles, probe tips as needles, and so on that's pushing a stick through a hole.
The one exception as I read it, are those that have used drills to clear holes they couldn't get the solder to melt. Unmolten solder = insufficient heat. Its really that simple. The causality can be more complicated (tip size, molten solder on the tip, and so on, that are all related to thermal transfer <get the iron's heat into the joint>).
I don't dismiss preheating entirely, but I would rather go for the minimalistic approach if possible. I don't think I have anything suitable handy, and it looks awkward for larger things, but I'll read up more and see how it works.
Assuming you've an appropriate tip size, have enough molten solder on the tip, sufficient flux, and possibly adding a lower melting point solder as previously discussed for best thermal transfer for the particular joint & equipment used, the proper approach is more heat.
This ^ is where I'm assuming you're at.
So adding heat is the minimalistic approach with this particular board with your equipment and techniques. Any other means without adding heat under these conditions, is risking damage to the board.