First off, you can't just jump to the conclusion that a particular problem is down to bad caps. It's true that they can and often do give problems but it's also true that this is often just being repeated as a mantra, in a cargo cult fashion, by those with no better understanding. But you have an ESR meter to hand, so you're up one on the game already.
What you need to understand is, it's often impossible, at least misleading, to measure such component values "in-situ", i.e., when they are mounted in the PCB and therefore affected by other components in their circuit path. There are meters which claim this capability but reality is a different country. In many cases you really do need to unmount them from the board (at least, lift one lead off) and test them in isolation against their data sheet values to be certain whether or not they are really the source of the issue. And definitely not all bad capacitors explode and/or show outward visible signs of failure.
If you're having trouble working out which leads belong to which components on the board when you're looking at it from the track side (and there is an art to it, gained with experience), it might help to photograph the top side and print it out, scaling it to the size of the board. You can then offer the print out to the bottom side, reversed, as a visual guide. Alternatively, depending on the thickness and material of construction of the board, you could also try shining a strong light through from the component side.