It was the biggest thing I was worried about too, Cobra, but it really works. You kind of snap to enemies when they're in a staggered state, which really works tactically given the speed of the combat, and the melee finishers are very brief. Most of them take maybe a single second, and if you're playing on a difficulty level that's actually challenging you, the health boost you get is welcome. It adds a nice tactical consideration to the moment-to-moment gameplay. Where I am in the game now, they're almost not even noticeable, as I have an upgrade which more or less doubles their speed, and another upgrade that makes me faster for 5~ seconds after doing one. When I'm going full tilt, the finishers are almost too fast to even see, and the focus is still very squarely on the gunplay. I understand the fears about them, but they honestly turned out to be a plus in my book rather than a negative. It gets heated enough that the melee animations become a welcome brief respite, and one you can't just do them willy-nilly without putting yourself at risk a lot of the time.
You CAN turn off the flashing indicator that lets you know when an enemy is staggered, and sometimes enemies will drop health and such even if you just shoot them. You can try to tailor the game to make as little use of the finishers as possible if you want to, by using upgrades that focus on other stuff, or you could even just play on a lower difficulty and completely ignore the melee button.
EDIT -
Tried out some multiplayer this morning. It's ... underwhelming. It's not BAD, it's just nothing at all like you would want out of a Doom multiplayer mode, which you would expect to emulate Quake on some level. The level designs are good, but the movement somehow seems too slow (no bunnyhopping), and it's overly reliant on armor pickups. If you can shoot a guy with 5 rockets and he isn't dead, something is wrong. Damage numbers are on by default, and max rocket damage is 55, which is anemic. I haven't unlocked all the guns yet, some of which appear to be exclusive to the multiplayer, and the few I do have feel kind of not great. Gauss for sniping is solid enough, rockets are at least okay, and super shotgun does the job okay, but having to pick a loadout of 2 weapons and not being able to change it up at all is a huge letdown.
I've only played a tiny bit as I don't have much interest in it, but so far, it's ... okay. I hate the level-up model of multiplayer, as I hate anything that hamstrings better players and elevates shitty ones. I couldn't even win a match for the first hour because nothing was unlocked, and siphon grenades almost appear to be essential to survive more than 10 seconds.
Tonally it's also all wrong. The taunts are fucking horrible, and while the armor customization stuff is neat, most of the armor looks kind of stupid, and you don't even see your custom weapon colors on your hud weapon, and everybody just has red or blue armor regardless since everything is team-based, so you don't really see people's custom stuff until the summary screen, when they're teabagging you or doing a stupid dance.
And yes, it's all team-based. There is no deathmatch. TDM, but no DM, which is mind-blowingly stupid and I hope they'll patch something in. I believe DM is an option in Snapmap, but that's entirely separated from the multiplayer proper, which is another weird thing: you literally have to load campaign, MP, and Snapmap as separate executables which require you to restart the game. Whoever thought that was acceptable is also nuts.
So yeah, MP isn't un-fun to spend time in. I more or less had a good time in the end, but it suffers from being the kind of MP that basically moved me out of the MP space to begin with—I have less than zero interest in the popular military-themed online shooters of today, and this is hitting all of those bases rather than appealing to people like me.
DOOM's MP also doesn't jive with the base game. The movement is the same, but the gameplay doesn't make use of it. All the moving and jumping and grappling doesn't translate to you living longer like it does in the single-player; instead it's almost more of a liability, as humans will effortlessly track your momentum when you jump, making it better to stay on the ground where you have more control. This is the antithesis of Quake-style MP dynamics. And granted, this is Doom and not Quake, but Quake is really the nearest analog to what they should have been striving for.
Anyway, it's not what you think of when you think of Doom or id, and due to the weird tonal issues, it can feel like everyone you're playing with is a 12 year-old boy, which makes it irritating. I highly doubt I'll be spending much time with it in the long term, and I'm very sad to see that this is what their season pass has focused on. I hope they'll realize that the vast majority of people will be playing this for the campaign, and hopefully offer us some expansion-style content for that in the future.
Multiplayer was farmed out to another company, which may explain some of the tonal problems, but it's odd that while Bethesda killed the original Doom 4 because it looked like "Call of Doom", they didn't do that in the MP space, where that problem obviously has lived on.