Why does my minecart stop moving on save/load?

Yes, this is a known bug: https://bugs.mojang.com/browse/MCPE-24859
The best you can do against this bug is to vote for it (requires you to create a bug tracker account).

As a workaround, you could make a minecart start/stop station where you park the minecart before you exit the game and which starts it again after you joined.
To do that, just put a powered rail at one end of the track (with a block behind it) and a lever next to it (or on the block). Before you exit the game, turn off the lever and wait for the minecart to arrive there. When you start the game again, turn the lever on and the minecart will continue driving.
This is not very nice, but it's at least a workaround until the bug is fixed.