Even with subtitles on you're going to miss a lot of informations, even if you see "Soldier damaged" or whatnot, you won't be able to deduce his positioning.
Also you wont hear steps and/or people shooting around you.

I can't think of a logical reason why that would be the case but you could in theory play with subtitles, the issue is subtitles don't indicate the direction from which the sound is coming from which is pretty important because knowing where your enemy is located is half the battle when coordinating a push or setting up a defense.

Edit: I got beaten to it :(

You are going to miss some things like, sniper shots, de-cloaking of a spy, general noise that players make that would alert you to there presence, the general atmosphere of tf2 and also the speckled purple robin that tweets occasionally. But sure.

Sounds play a larger role than most people expect, It's usually noticeable after a little while of playing without it. Your brain will catch on to shots being fired in the vicinity behind you, hearing your teammates call for medic or thanks somewhere on the map, uber calls, bombs, and so on. I reccomend you work on improving while keeping sound on.

Have you tried playing with "volume .01" or a volume just low enough to barely hear what's necessary (personally I don't cound footsteps as being necessary more like Gun Shots, medic calling uber, and decloaks are definitely important though)? having really low sounds might help if you get distracted easily while also still being able to distinguish things going on around you.

you can turn your sound off if you're willing to trade sound for "dm" as you put it. if you don't play seriously, there's little harm in it if you're willing to accept the consequences of doing it. if you want to be really good at the game, you'll need to have sound to listen to soldiers jumping, stickies being placed, and even foot steps.

i will say that i think you're experiencing a placebo with better dm while you have no sound on.

I personally use
It's just loud enough to hear everything necessary, but not that it get's in the way of hearing comms or yourself think.

I keep my game sound relatively quiet and keep Mumble fairly loud over m game. I play mostly Highlander, so there's there's lot of white noise. It's still important to have sound regardless.

If you are going to turn off sound I'd recommend trying this, I find it much easier to recognize how much damage I did by my hitsound's pitch than having to look over and read the number.

http://www.teamfortress.tv/post/490552/bounty-mute-all-sounds-but-hitsounds