Dont use IC sockets; they are not necessary.
If the circuit is properly designed, meaning that all IC datasheet parameters are met, you should never see an IC fail. I've built many nixie clocks, all of which I designed, and I've never had an IC fail on it's own. The first failure I've had was a few weeks ago, caused by ESD (my fault) on a Raspberry Pi, so technically not one of my designs.
IC's are actually easy to remove without damaging the PCB, though you will likely destroy the IC in the process. For thru-hole IC, flow fresh solder into each pin, and while it's still molten, use a vacuum desoldering tool like the Edsyn Soldapullt (see below). I've been using mine for ~40 years now, and it still works great. Once the pin is sucked-out, wiggle it with the soldering iron and it should be free. Once all pins are cleared, you should be able to gently wiggle the IC out with your fingers. If not, snip the stuck pins off the IC with fine clippers, then use tweezers to remove the pin while heating it. After all pins are clear, mop-up with some solder wick, then clean with 90% isopropyl alcohol. PCB should be just-like-new.
SMT parts need careful heating with a hot-air reflow tool, and you should be able to gently lift it with tweezers, though some pins will get bent and most likely the IC has been thermally overstressed, so toss it away. Clean-up is same as above.
Lastly, I never breadboard a design. PCBs are easy and cheap to have fabbed, and have far superior quality. If you mangle a board with debugging, build another. But if you do proper simulation beforehand designing the PCB, you wont need to do any debugging.