Should I replace the crossover capacitors in my vintage AR 3a speakers?

Indeed, specifically old elctrolytics. Film to film is usually OK.

What I find amusing is when modders replace old electrolytics wiht film caps, removing 0.5 Ohms of ESR or more in the process and then proclaim what a difference it makes. Well, yeah, it kind of has to at that point, for good or ill. Same with changint DCR in coils without understanding the whole circuit.  When I make speaker crossovers I specifically pick among the various gauge coils to get a specific final sound.  I don't get small gauge coils for cost savings.

Not saying good caps don’t make a difference. Just saying it’s good to know what’s changing.

OP:

Honestly the issue I brought up is rare, but when it happens it really messes with people who don't understand why their new fancy caps made the speaker severely change character. 

It's worth doing a full impedance simulation for this reason.  Also helps you understand if your changes are due to changes in ESR or actually better sounding parts. 

Generally, 1990s electrolytics and before were a lot higher in ESR than today's versions, so that's also something to think about when renewing old speakers.

Erik

Only as a learning exercise.

Be careful replacing parts that are in parallel to the driver as the old parts may have ESR that is part of the design. That’s actually true for all caps and coils but the one’s in parallel can cause unexpected impedance swings if not kept.

If you go down this path I strongly encourage you to get an impedance test tool like this one from Dayton (as used by some mega brands) or you can build a free one and use Room EQ Wizard. The benefits of this vs. a single cap/coil tester is you use it to test the entire speaker as well as individual components. It will measure uF/mH as well as ESR/DCR. Measure the speakers before hand, and after to make sure all is well.

If things don’t work it’s an incredibly good diagnostic tool. It’s kind of easy to tell what broke, also you can plug the data into a tool like XSim or VituixCAD to do a full electrical simulation.