A new tool I have been considering is a real capacitor tester ... my multimeter can only test capacitance up to something like 20uF
That's about what I was thinking when I really got going with vintage audio gear repair. Here is what I observed:
For vintage tube gear where you have high voltage (e.g. 125-800 VDC) needs or oddball capacitance values (e.g. 5 uF), there are times when you may want to keep the original cap unless it is already bad. A cap tester that 1) includes leakage testing (which must be done AT the rated voltage) and 2) has a high test voltage range (e.g. 3VDC to 400 VDC) is the only practical device to do this. That means a VINTAGE cap tester like the crazy-expensive-now Heathkit IT-11. I've bought, restored and used these... only to realize I don't LIKE relying on 60 year old caps. I now automatically buy replacement caps for all electrolytic and tantalum caps and often buy film and ceramic replacements. Yep, that's the shotgunning approach. My time is valuable, I don't want to go in a second time, and the time it takes to test each cap is better spent (for me) outright replacing them.
For modern, post-tube era cap checking, you could certainly get a nice tester like Sencore's LC line. Very accurate, great gear, and very expensive even used. That will tell you 1) if the cap is working properly and 2) what capacitance and ESR value it currently has. Good to know if you are choosing which caps to replace and which caps to keep. This device supports the non-shotgunning or "replace only when failed" approach. Perfectly valid, just takes time.
The $20 cheap component testers off eBay and Amazon and banggood really work. They accurately tell you capacitance above 200 pF and below 10,000 uF, if I remeber correctly. They also tell you ESR, I think. And they do five other critical measurements for repair, restore, and recap. These are invaluable, mostly accurate, cheap tools. I have three just because they are so cheap.
Peak Atlas also offers ESR and capacitance meters. Not TOO expensive, but still five times the cost of a cheaper, more versatile $20 cheap component tester. I have the Peak Atlas DCA75 pro something or other to get precise answers about diodes and transistors. It does curve tracing very nicely. Their gear is small, well documented and modern. But I'd buy a $20 cheap component tester long before spending the extra on a Peak Atlas device.
So, long winded blah blah blah. But some observations and opinion for you to consider as you like.