Leo Fenders amplifiers are probably the best known for this feature.
After all, guitar and bass amplifiers saw them first in his excellent amplifier designs.
Leo was not a trained technician as such, but rather, self-taught at a time when there were few books or guides available.
So he learnt from RCA service manuals written by clever technicians, that using a standby / power arrangement was the correct method.
Leo was originally a radio repair man in an era when radio was the form of communication. Radio transmitters used some very powerful valves (tubes) and best industry practice introduced protocols for start-up and shut-down.
Standard Operating Procedure and Protocols are typical in any industry.
Perhaps by analogy, we could consider similar protocols we use daily without thinking much about them, such as starting or shutting down the computers at home or at work.
Therefore, it was entirely logical and best practise to Leo whilst designing his amplifiers to include a similar start up and shut down procedure, as was current best industry practise during that era.
The salient thing to note is that the RCA technicians from whose service manuals Leo learnt, were not building small guitar amps, which used similar valves to large domestic radio sets of the day, but huge radio transmitter stations, so their protocols were not directly equivalent to Leo's application.
The same service manuals explained that leaving equipment in standby should not occur for long periods, that is harmful to the electronics.
Three to five minutes at start up, and simply turn the volume down if you have a short break in a live set rather than use the standby switch, which would otherwise make a noise.
Strictly speaking they are not necessary for guitar amplifier applications, but as more powerful amplifiers were introduced involving higher voltages, using this procedure aided correct loading to the relatively inexpensive capacitors Leo used, to keep costs down.
They don’t protect the valves, which most people seem to believe, but by disconnecting the high voltage allow the cheaper capacitors to fully actuate before introducing full power to the valves. It will not extend the life of the valves, but can reduce their life. Valves don’t mind if heaters and high tension occur simultaneously, but overall helps the circuitry.
Interestingly, the facility to switch off high tension is a useful tool in research and development situations. Especially when a lot of tweaking of circuits is going on for instance in development stages, or when there is a fault and diagnosis necessary. Its far quicker than constantly waiting for things to warm up again and again.
So, useful if you are designing amplifiers or repairing them like Leo Fender.
Don’t worry too much about it.
Use the standby to warm up for a few minutes.
Also use it for short periods if plugging in or out of the amp to reduce popping if you want.
Don’t use it to keep everything warm while you have a fifteen-minute break, turn the volume down instead.
Personally, I tend to use the standby switch getting the amp warmed and cooled at start up and shutdown, and allow plenty of cooling time before anything gets moved. Heat is a real issue with electronics as is constantly switching on and off.