'I want my starters in a close game at the end'
Your own sentence. That's where the the upgrading the bench player is the solution. If your bench player is solid, then the substitution will not be in the end of the quarter. Thus resolving the "bench finishing the game issue".
You are more of a highlights NBA watcher? I own a league pass and I have watched 82 dallas last season games and a bunch of good matchups (heat vs lakers etc) + all playoff games and finals live. Mostly (there are exceptions though) the bench starts the 4-th quarter, thus letting the starters rest and finish the game. Surely I should not have to explain this to an american.