Plumbing – What would cause the shower water to start hot and then, after a minute, settle on warm water

I think you are right. There is a minimum flow rate required before the boiler will actually kick on. I have exactly the same thing with my tankless hot water heater.

Aside from getting a separate "point of use" hot water heater for just the sink, The only way I'm aware of to get around it is to turn the water all the way over to hot and on full blast. That usually gets enough flow going to start up the water heater. Once hot water starts coming out, you can adjust the temp a bit (at which point the water heater will probably shut off, and all you have is the hot water left in the pipes).

Your downstairs shower has a restriction of some sort in the water supply. Start at the showerhead and work backwards to find it. You need to check pressure while flow is happening - a restriction will allow full pressure when there's no flow, but will cause a large drop in pressure when there is flow.

For the easy possibilities, check any supply valves to be sure they are fully open, and check that you don't have a limed-up or otherwise overly restrictive showerhead. Other issues can be much more involved to find/resolve - my classic warhorse of a tale is the shower/tub with terrible flow on the hot water side, due (found after I reached the point of disassembling pipes, because something was clearly very wrong with them) to one elbow being 95% full of solder, leaving only a tiny hole for the water to pass through.