Plumbing – No hot water until another faucet is turned on

No other fixtures are "dependent" on the shower valve. What may be happening is what is called "crossover", hot water crossing over into the cold water line (or vise-versa) due to a defective single-handle mixing control. It sounds like your plumber may have been right, as this would tend to not have a noticeable effect when the shower is flowing. When the shower is static, you turn the hot on at a different faucet, cold water is pulled through the defective shower cartridge to mix with the hot in the supply line, resulting in luke warm water.

Troubleshoot by positively shutting water supply to the hot water side of your system and checking for flow at a hot water tap, or have a plumber replace the shower mixing valve

You just replaced the tank so it shouldn't be a broken dip tube as I had first assumed. My next thought is there might be a blockage somewhere, but you should notice that as a loss of total pressure (compare the flow rates of your hot and cold coming out of a faucet somewhere).

You should probably remove your shower valve's cartridges and clean them while you're purging the entire system and flushing out debris.

I have a single handle shower valve that doesn't get as hot, or for as long as I'd wish, so I have cut back the cold water by closing its supply valve a little bit. This adjustment alone has increased my acceptable temperature dilation from ten minutes to upwards of half an hour.