If it's average consumer-grade stuff, yes, I'd get out the meter and check the electrolytics. But I'm not going to junk a bunch of oil capacitors because they're "too old". And there's a whole bunch of fairly recent stuff that's been destroyed by "shotgun replacement" of caps. Any time that soldering iron comes out, one must take into account the possibility of damage.
It's the "shotgun" advocates that I have the biggest problem with.
So you do know old electrolytics blow stuff up, then? OP is asking about electrolytics; specifically, low ESR stuff in semi-modern power supplies. Things that are known to have problems, due to a number of issues ranging from growing pains to counterfeits. Where doing nothing gets you boards like I provided pictures of.
I have seven 90s pick-and-place machines that have required full recaps in their AC servo drives. Nothing else is wrong with the drives, but if you run them like that, resetting or locking out error conditions, eventually you blow up power electronics in them. You can still get them from machine rebuilders for around $1000/ea.
The controllers for those servo drives have 90s surface mount capacitors on them, under conformal coating. It helps to hold the leaking electrolyte in. Every single one we opened had at least one leaking surface mount electrolytic.
If I open a power supply with any real age on it for basically any reason, it now gets a recap. If I get some vintage test equipment in, it gets a recap before I bother to calibrate it. Tube equipment gets a recap not only for electrolytics (which all go, no matter what) but also wax paper capacitors and anything that looks like a wax paper capacitor. I don't care if it *might* be a di-film or if some guy's tester with a 9V battery says it's OK, or even if the HV Heathkit cap popper says it's OK. It's not worth blowing up something nice for $10, $20, or even $50 in capacitors. Does that make me a "shotgun advocate?"
W.R.T. stuff being destroyed by bulk recapping, I would argue that the kind of folks who butcher a cap job would've probably wrecked it anyway.