1/2″ Feed, 3/4″ Piping, Effect on Water Flow Rate

Currently plumbing a 20" rain shower (droplets + water fall). Using it with a water pump/booster.

The valve that comes with it has two 1/2" inlet which consists of cold + hot water so it's basically a 3/4" feed.

It however only has one 1/2" outlet.

So my question is does it help if i connect 3/4" pipe to that outlet after using reducer coupling? In this sequence:

Outlet > 1ft of 1/2" pipe > 10ft of 3/4" pipe > Rain Shower

I know bigger pipe helps with friction but it's of no concern here as the distance is short.

My concern is more on whether this arrangement will produce a 3/4" water flow instead of a 1/2" water flow and whether i should just go fully 1/2" in order to avoid wasting hot water that are sitting in the pipe since the flow of the said arrangement is just going to be the same as a 1/2" flow.

Thanks.