The other thing about using Price in this game is that if he wants to straight up pitch Stroman ahead of Price in Game 5 (which I would like to see), using Price here avoids having to start Stroman ahead of Price on shorter rest, which would be a pretty clear "you're not the #1 here now" message to Price and could really wreck Price mentally. This way, with Price coming into the game now, Price gets a bit of confidence (or he shows you he's a true playoff choker when you still have a chance to recover), gets set up for Game 1 of the next series (if it happens), and he gets to use Stroman in Game 5 without upsetting the natural pecking order.
Also, where I wouldn't want to pull Price from the game after 4.2 innings because he's obviously got playoff headgame/mental demons, I think this is more likely to motivate Dickey than to screw him up for his next start like it would Price.
Again, I don't really like these moves, but I can see how if it works out he gets Price/Stroman/Estrada/Dickey set up for the next series in their "natural" order without having had to publicly drop Price down a peg by starting Stroman ahead of him.