There sure are some interesting answers and "opinions" stated as "fact".
In fact, it's not nearly as rigid as it appears, and Leo Fender did not include standby switches because he didn't know what he (or his engineer) were doing, or that they were being cheap, or many of the rumors that seem to permeate such threads.
Here's how I see it, based on being inside the industry and having designed tube amps for commercial applications that had to be reliable and convenient to use:
1. Are standby switches necessary with tube rectifiers? Generally with tube rectifiers they are not necessary for reliability because any (reasonable) possibility of cathode stripping is addressed by the warm-up time of the rectifier tube and the soft conduction of the rectifies as it warms up and begins to conduct. I went back and looked at me Ampex (not Ampeg) service manuals to check that I was remembering things correctly, and many of the earlier tube amps (used in cinema and large scale public address) did not use standby switches. Most of these applications were of a higher level design than MI amps, so voltages and stresses were accounted for.
2. Are standby switches necessary with solid state rectifiers? Here's where the gray area begins to rear its head... if the voltages on the plates and screens are "reasonable" there's probably not much of a penalty for not using a standby switch, BUT as the voltages get higher there may be reliability benefits for using them. Also, the more an amp is turned on and off, the more likely that any damage would be cumulative. In the Ampex cinema amps that used solid state rectifiers but did not use standby switches, the voltages were generally more reasonable than many MI amps, and they are generally turned on and off a much smaller number of times than a guitar/bass amp. With some of the very high voltage MI applications, I believe that there are significant benefits because the amps are (or would be) generally turned off much more often than commercial applications.
3. Do standby switches protect capacitors? I don't think so because just as often the filter capacitors are located before the standby switch. The open circuit voltage of the supply is too high for the capacitors, that's a design issue not a capacitor or voltage issue. Especially these days, but it was true 50 years ago as well, the lifespan profile of an electrolytic capacitor is fairly easy to calculate.
4. Why may they be beneficial and why they are included (my opinion)? Because players often turn an amp on and off many times during a single days use. The cumulative damage may in fact accelerate the aging of a tube, but leaving the amp on (without standby) during that time would generate additional heat which accelerates aging of both the tubes and electrolytic capacitors. Since many of Fender's designs were with the tubes mounted "base up", this means that the heat rises and the chassis temperature increases. In standby, the power dissipated by the amp drops by roughly 50% which lowers the temperature of the chassis and decreases aging.
5. I think many players find it more convenient to place the amp into standby while fiddling on stage without waiting for warm-up when switching to operate.
6. The dreaded "vintage line voltage" issue... this is nowhere near as common as some folks would have you think. Going back through my service data, most vintage amps are designed for between 115 and 120V, with some tolerance for under/over-voltage. Today, the designs are even more refined in this regard because the safety regulations require testing at +/-10% of the nominal line voltage. These values are used when selecting components minimum acceptable voltage ratings. There are some exceptions, but there's much less in reality than the mythology might suggest.
So yes, standby switches are provided for good reasons, they are not provided because the manufacturer it trying to be cheap (in fact good standby switches are quite expensive due to the required voltage rating).