My dedicated ESR Meter (the CapAnalyzer 88A) is used every day on monitors and power supplies. I don't trust those "multi-function" ones that you describe as testing transistors and diodes. I like dedicated ESR meters myself, but it's up to you.
If I'm working on a Wells Gardner 4600, 4900, or K7000, for example, and it's never had a capkit, I don't even bother testing caps. I just go right ahead and recap it automatically.
Now if I'm working on an oddball monitor or a fairly new one, then I will use the ESR meter.