Don't they all do that?
Select your start and end points; select the primary route; use "destinations" or "way-points" to adjust the route.
The only catch I have found is that some (most?) apps place a restriction on the number of way-points, like a max of ten.
Yes, you are right, Google Maps app does not give you that option. You need to create the route via the web site.
Cannot seem to get Waze to do manual route adjustment either. :-(
I usually end up using Sygic. I liked it well enough to buy it.