Those transistors that failed to short, will activate classic continuity beep. Beep, OK that one's gone. Same as any discrete semiconductor.
Also like xavier60 says, in bridge, rarely they don't blow in pairs, but if it isn't cost prohibitive, I would change them in pairs even if one of fets looks OK.
It probably spent some time outside safe specs and who know if all is OK.
Like Martin says, I will disconnect/pull out transistor and check it with tester to see. Even that is not real life test, but at least it is in isolation and you can check basic parameters. Once I had soft starter for motors with large monolithic 3 phase FET bridge, that I improvised a jig that was testing it with 10A to see if it works ok. Small transistor tester from Atlas was showing nonsense...
If I had to check 1000 diodes a day, I would definitely think about if some specialized diode testing would do me good.
As I said, you should use any advantage that helps you do your job better and/or faster, depending on what is important.
I usually don't have to do things superfast, but it has to be right first time.
If you're changing 400€ 3phase MOSFET bridge on high power inverter, you won't plug it in to check if it's going to work now. You will check and measure 3-4 times and double check everything before power on. Sawing few minutes is not going to be priority..
So, yeah, like I keep repeating, it all depends on what you do, and what your methods are..
For my work, I simply use different troubleshooting technique, that doesn't benefit from diode beep mode, and it is not optimised for 100 transistors per hour but for something else. I understand other people have different priorities..