EE also, designer of VFDs at the last employer.
1) Most VFDs are N.O.T. designed for component level repair.
2) dead IGBTs almost always means dead driver chips. Count on replacing them just because.... don't even test unless you replace them... I've repaired enough VFDs in development for various causes to see that. The shorted IGBT usually melts a spot on the die, which makes a gate connection, shorting V+ to gate (and driver).
3) Sometimes the failure "reaches back" through the driver to damage the control uP, or burns holes in the board, in which case you have a "problem".
You can check them in operation, so long as you have a 'scope that has isolated inputs. HP has made portables that do, and Fluke Scopemeters do also, most isolated up to 600VAC.
It's a very good policy to check gate waveforms, to make sure that there is good drive, with a sharp rise to the "charging plateau", and then a rise after that to well above the "plateau". Slow gate drive will cause another failure. You want 100 MHz to see that.
Overall, it is actually pretty reasonable to replace entire PWBs, or drives, and not parts. You can have more confidence in a unit that has not already failed and been repaired, especially since newer SMT parts are a real nuisance to replace. Many of them have the terminals underneath, inaccessible. You need a hot air soldering wand for those, and reheating boards increases failure rate.
If you can get a deal on a bunch of drives of one model, it may be worth it, if they all do not have the same problem. You can often mix and match PWBs to get working units. (watch out for potentially incompatible revision levels).
You can, but do not have to, match VFD output current to motor current (VFD a bit higher). If the discrepancy is too high, however, then the VFD may not have sufficiently fine control of current etc at the motor current level, so that you cannot protect the motor.