Yes, capacitors do age and heat will accelerate the process. The electrolyte that's between the sheets of foil that make a capacitor work will dry out and can cause the capacitor to short out which can take down other components. Bringing the voltage up slowly can reduce the damage when one fails, and can help 'heal' some caps that aren't quite dead yet. Replacement with quality parts is always the best option going forward. Would you drive on 20+ year-old tires?
A lot of Digital Volt Meters, DVM, have a capacitor test function so you can check components once they are removed from the circuit to see if they meet spec. While obtaining replacements and soldering them in isn't trivial, it's not really tough. Access is probably the most difficult thing as old tube chassis can be quite a tangle of wires and components. The lovely linear layout of modern electronics hadn't become the norm and most tube-era wiring is done point to point. Resistors don't usually need replacement unless they look like they got hot at some point. They will look dark or burned usually.
My most dramatic tube equipment repair was the transmitter in the student radio station at the University of Missouri in Columbia, KCOU 88.1FM. It's relatively a low power transmitter at the top of one of the dorms outputting 225 watts ERP. Modern transmitters are all solid-state transistorized units. This transmitter had been in operation since the late 60's with tube technology that would not be out of place in the '30s. Big tuning coils, large capacitors and resistors and a four-kilovolt power supply driving a single, metal encased with mushroom-like cooling fins, radio frequency power output tube.
I was taking the FCC-required readings at the transmitter at 11pm on a Friday night and when I flipped another switch on the front panel to read the voltage on the tube grid there was a bang and all the needles fell to zero. Uh Oh. I called the DJ at the studio and told him to go home and lock the station, we weren't going back on the air that night. To make a long story short, I eventually figured out that the tube had reached the end of its life and my flipping the switch was just enough disturbance, either electrical or mechanical vibration, to cause the tube to short internally and blow out a large wire wound resistor that handled some of the 4KV that powered the tube. The large oil-filled capacitors in the circuit survived the voltage spike. With a new tube and new resistor, it was back in operation.
Consumer tube equipment doesn't have that kind of power, but it's not trivial. Bad caps can fail dramatically. Unless the components are exposed and their patina is important for visual appeal, I'd replace all the caps if I was going to regularly use tube equipment.