I have been playing DOOM (2016) lately and it's pretty great, however I am quickly getting tired of looking for secrets and all kinds of collectibles, spending upgrade points of all sorts, etc.
I'd just want to cut out the awesome, fast combat and just rock llike that for hours. I remember whole-night parties, playing Unreal Tournament - such a great, fast-paced and brain-less fun.
Another example can be Max Payne and Max Payne 2 which I played lately - great shooting, no unnecessary exploration, just pick up ammo here and there and I roll out again.. plus awesome story in both.

Can you recommend me a game like this - fast-paced, without tons of collectibles (which seem to be mandatory in most modern games..) or other weird distractions?
It does not necessarily be a shooter, can be a melee combat or something else. Thanks in advance!

Painkiller might be a solid FPS for you. It doesn't seem to have a ton of collecting but the action can be definitely intense.
"Legacy Of Kain: Soul Blazer" is a fantastic action game.

I love "Arx Fatalis" for its exploring but you might not want to collect things so I'm left with you playing fighting games or shmups.

Well, any id game should be perfect, like the Quake series or Wolfenstein (return, tno).
Among shooters, Necrovision comes to mind for its relentless action and brutality.

Rune is an action game you might like, but be sure to play it on normal, as in hard enemies become bullet sponges and transform an aesome, gory and fast paced game in a damage sponge fest.

If you don't mind a platformer, I'd say Slain! Back from Hell.

If memory serves, Vanquish is supposed to be a fun, fast-paced third-person action/shooter. I've not played the whole game myself but I tried the demo once and recall it being quite a wild experience at certain encounters. It was developed by Platinum Games so I think it's safe to say the action is fun and over-the-top. They also made Bayonetta and and Metal Gear Rising: Revengeance and I don't know if they have collectables but they are generally considered to be very fast and fun games. Hope that helps!

Shadow warrior, mainly I suggest this for the reboot one which is great and just what your after, and less so the newest one which is still good, but not up to the reboot one. Also the original! And add to that blood 1 and 2.
Then there is Fear 1 and 2.
Far cry 1, that is another cracking shooter. 2 looks nice but is very boring. 3, and 4 are not here, but well worth it, so is primal.
Ion maiden is shortly coming.
Daikatana if you like hard.

Hell there are thousands!

Serious Sam games sound like that. Sure there are secrets too but they are totally optional, usually just meaning you can get some more powerful weapon earlier than normally, or some easter egg. And quite often the secrets are quite obvious.

The main point of those games is shoot shoot shoot, ballet with big guns. One of the most enjoyable FPS games for me.

As an UT lover (and UT2015 being RIP) I like Overwatch in arcade "random pick" mode. As a fun single-player I can recommend Carmageddon Max Damage although that also runs in LAN mode.

Republic Commando is a great FPS. Very intense, but without beeing smash-you-keyboard hard. It has a tactical element to it, commanding the other three guys in your squad, but it's very simple and doesn't slow down the action in the slightest.

Another great choice would be Bulletstorm, the greatest FPS ever in terms of just sheer balls-to-the-wall crazy action, but unfortunately it's not available on GOG :P

Xotic - http://store.steampowered.com/app/92600/Xotic/
(it is bart of Fanatical FPS Heroes 2.0 Bundle - https://www.fanatical.com/en/bundle/fps-heroes-bundle)

Just Cause 1,2 and 3

This might sounds weird, but Dota 2?

I saw my brother played it and it seems like non stop section with little management (get money, buy upgrade etc).

Arpg (Diablo and the likes) sounds suitable for you. Twin stick shooter genre too.

I'll go out on a limb and suggest Tetris: The Grand Master and its sequels. Yes, it's Tetris, but it's very fast Tetris, No exploration, no distractions, just superfast block stacking.

As for playing them:
TGM1 and TGM2 apparently work in MAME. There's a modded version of MAME out there that's supposed to reduce input lag (which does become an issue in high-speed play.).
There exist clones (like Texmaster) that you could play instead of the originals.
Of course, if you have the money you could try to acquire the original arcade PCBs and play the games that way, but this can get expensive.

I believe it's recommended to start with TGM1 first and then later move to TGM2.

I woul've said Blood.
https://youtu.be/PH2lLD6wtz0?t=20