Shower – In a low water pressure area, why would water from a shower take several seconds to flow after the shower is turned on

The stopper you're describing is called the diverter. The symptoms you're describing sound like it's gotten slightly bent or nicked. I would take it apart and try re-setting the pin. Maybe scrub it a bit with a light abrasive to true it up. If that doesn't work, replace it. I assume your diverter is in your tub spout. You can usually remove that by removing the hex-head set-screw and then rotating the entire spout (the padded handle of a hammer stuck in the spout can help you get a grip).

If the cold water pressure is good, then you are right about the shower head not being the problem.

Check the hot water shut off valve feeding the tub to assure it is opened fully.

Also if you have good hot water pressure at the vanity and are sure shut off valve is fully open, the problem may be in the mixing valve that feeds the shower head. It's not uncommon for deposits to form over a number of years in this valve and restrict flow, especially on the hot side. Could also be a broken or stuck washer in the mixer. You can shut off the water to the valve, hot and cold, remove and inspect the cartridge from the front. Remove the handle(s), the bezel trim and the valve cartridge should come out with a wrench. If any of the internal parts look suspect, take it to your hardware store and they can help you select new soft parts to rebuild it. Replace it in opposite order that you disassembled it, and don't forget to put some plumbers putty under the top and sides (not very bottom) of the bezel, so water won't leak in behind it and down your wall.

Good luck.