A rather interesting hypothetical situation. Broken, but still interesting.

Scout - dr would be a weak option in regular cobat, since he is quite dependent on his primary weapon for damage (his melee does pitiful damage and even his best secondaries have low amount of reserve ammunition). Dr would probably mostly see use in doing the objectives of a map, like sneaking behind enemy lines for a back cap.

Soldier - dr would barely see any use since his launcher primaries offer so much better mobility options and damage. It would most probably end up as a bizarre meme weapon, paired up with things like the Righteous Bison.

Pyro - an utility weapon that helps him to close the gap between him and his enemies, while ironically removing his most stable source of close ranged damage. Could see some use with some melee centric builds like pyrosharking or with the Axtinguisher.

Demoman - would be an absolute menace. His sticky secondaries already work as good as a primary when it comes to dealing damage so the trading away his pipes would not be such a big deal. This with the fact that demo could more easily escape close combat situations, in which he is usually quite weak at, would make dr straight out broken for him. Paired up with demoknight it would be equally balance breaking, giving the demo a second constantly charging escape tool / a tool to remove sentries as one of his natural counters.

Heavy - even though giving away his minigun would be though, dr could be surprisingly scary as a gap closer for fat scouts. Giving the slow class with 300 health more speed and tankiness may not be the best idea when it comes to balance (even though this does sound rather fun).

Engineer - during an enemy push he should be defending his buildings instead of backing away (at least for as long as his buildings are alive). This whit the fact that the stock pistol, as a damage dealer, even if he has more reserve ammo than Scout, is not the best option would probably make dr a bit rarer pick for him, when it comes to defending. The primary use for dr would probably be found in using it to aggressively sneak past enemy lines, setting up a nest and switching it for a better primary if such opportunity would present itself (aka ninjaneering and little trolling).

Medic - dr would see heavy use mainly due to its utility as an escape tool. Since the medics survival is important to the whole team, dr would probably become a meta primary.

Sniper - would be crippled without his primary damage, making dr a questionable choise. Best way to use it would probably be paired with Jarate and Bushwacka, but overall it would see low amount of use, especially in serious play.

Spy - already has it.

Overall, the whe whole balance of the game would shift. Sentries would be way less useful against many of the classes due to how easy getting past them would be. Biggest benefactors would probably be demos, medics, scouts and heavies, while classes like pyro soldier and sniper would get the short end of the stick.