I'd be willing to pay the plumber to put in a couple extra hours and put in a couple more pipe bends, if it meant not getting splashed with cold water every day.
You would need a circulation pump and a bypass valve on the sink in the bathroom or on the water heater. There still would be a very short period(time depends on pipe layout) of cold water unless you tear out the shower and put the bypass valve at the shower supply.
You wouldn't even need a plumber if you feel comfortable screwing lines together, but you might need an electrician if you can't add an outlet.
Ah, I misunderstood your point but it makes more sense now.
That would cost thousands and would require remodeling the shower. Not so bad if your tile is still in production or you can save the insert, but an older bathroom and you are going to be in it for thousands. Best case is you have access via the opposite walls and then you are in the low thousands unless you can do all the work yourself, but the end result will be kind of rough.
That sort of change is best saved for a remodel or fresh built.
Yes, the idea would be to stop building showers in such a silly way, and build them in a way that makes sense. As I said above, we worked it out with sink taps a while back, one day I hope it will be more standard for showers.