Should I recap a working Tek 453?

The scope is full of electrolytics, most of which will have been made by Sprague -- 30Ds. As Tom said, if it's working, it doesn't need fixing. Most failures will be obvious and fortunately, will cause fairly limited harm to other parts. Use 'til it don't work, then fix it. Great scope as long as your eyes are good. BTW, the 453 was designed specifically for IBM, for field service, then became a fave with everyone. They traveled well.
If it ain't broken don't fix it.

Usually I do agree with this but not in this case.
Electrolytic capacitors are the N°1 failure prone electronic parts.
Sooner or later they WILL fail. In a 45 year old instrument they will fail rather sooner or maybe even "soon" naturally.

I do own quite a lot of wonderful older/vintage test equipment from Tektronix, HP, Rhode & Schwarz, Philips, Brüel & Kjaer etc., most in the range from 20 to 50 years old and which BTW I just love for their partly absolutely umatched and uncompromised build quality. This stuff was built to last, if possible forever, not to break shortly after the warranty period ran out. Modern equivalents just can`t rival this quality anymore IMO (besides this I couldn`t afford those anyway).
Although most of them still do work more or less fine, I have found a few dry or almost dried out electrolytics in some of them.

Most electrolytics sit in the power-supply circuits, only few of them in subsequent circuits. Failure in a power-supply doesn`t mean necessarily that an instrument just quits working. It can do much more harm as this could damage other parts which might be much more difficult (or even impossible) or expensive to source and to replace (special precision resistors, hard to find obsolete semiconductors etc. etc.).

Modern electrolytics are good, dirt cheap, easy to get everywhere in any quality, value, voltage rating and size and they are small.
The latter is a good thing as there will be no problem with space restrictions and even the next higher voltage rating than the specified one will mostly fit easily.
The only exception regarding availability are the axial leaded type elkos which are in many of the older gear and which sometimes are already hard to get today as they don`t seem to be manufactured anymore. This is just another reason to replace (or stock up) those NOW before they are all gone and really hard to find.

Therefore only recently I decided to replace ALL of them in any of my older (>30 years) vintage instruments before some bigger failure may happen and that I´ll regret not having done this earlier.
I don`t see any reason why I should wait for a possibly major damage to happen when I can prevent this now very likely (of course other not capacitor related electronic failures can happen anytime, although from experience I can say that this is much more unlikely).
First I thought this will be a huge task. Now that I`m at it and that I have opened quite a few instruments and studied the Service-Manuals already to prepare lists of what I`d need, I`m surprised that in many of them there are only a relatively small amount to replace actually.
Same experience here: I prefer replacing elcos in time, before they fail completely and take a few other parts with them.

Sometimes even the performance of the gear suffers from old caps, recently I recapped a HP signal generator that was prone to unstability. With new caps stable as a rock.

See it like this: if the gear worked before you recapped, it will work after it too (if you did your job properly).

I have now a vintage turntable here for which I didn't find the time to replace the elcos before it stopped working. If I recap it now, will it work again? Or do I have to spend a few weekends to fix it (if possible at all, it's a really sophisticated japanese beast with lots of custom ICs).
You can check electrolytics by looking for bulging or leaking of fluid. If possible, disconnect at one end and check leakage current and capacitance value. If OK leave them be. They are unlikely to suddenly fail.

You can check other caps by checking circuit voltages, as leaky caps will disturb DC conditions.

Repair what is broken, or about to fail. Simply changing all caps is more likely to introduce faults than reduce them.