As others said, sometimes none of the old capacitors are actually bad, or if they are, they're failing in ways which don't really affect the sound.
For the past few weeks I've been listening to a Realistic Stereo 40A amplifier, from 1961, and all of its capacitors are original.. I re-formed and checked them all, and it sounds just fantastic! I would anticipate a re-cap to be very anticlimactic with this piece.
Usually when I do a re-cap, it's because I encounter one or more failed components, and understand the other similar parts cannot be far behind.
As for hardware that ages and is responsible for deteriorated sound?
-capacitors can get leaky or change in value
-tubes can get weak, go noisy, or develop heater to cathode shorts
-resistors can go noisy or drift, especially carbon composition ones
-plugs and sockets can corrode, and make poor contact or lose spring tensiion
-speakers can deteriorate in terms of weakened suspensions, changed resonant frequencies, dried out glues, etc.
-transistors can develop tin whiskers, or can go noisy
-controls can become noisy or dirty,
-switch contacts can wear out
-belts can stretch, idler wheels can flat spot or go hard
-anything with moving parts can suffer from old lubricants getting sticky or drying out
-old ears can lose high frequency response and sensitivity!