Roguelite, not roguelike. They're a lighter version. And it's mostly random,but not tunnels in this case . Just buy it, trust me. Nobody who's ever bought a game I recommended ever regretted it .

Heh.. you don't lack self-esteem do you? =)

Nope. Never had that problem .

Well that was obvious. =)

Edit: and it is a good thing.

I just tried Hammerwatch today for the first time. I don't know if it ticks all your boxes, but it's reasonably fast paced, there are plenty of traps and projectiles to avoid, you can go save your progress at a checkpoint anytime (which tend to be fairly close by as the levels are smallish and have multiple checkpoints). Death is permanent, but you can play with extra lives enabled (the default). No dialogue, cut scenes or other yada-yada. Great game? I don't know, but it's really simple and easy to get into and doesn't require much patience. Just lots and lots of shooting or hacking & slashing. Mindless action, if you will.

Volgarr the Viking
No saves, but there is a permanent checkpoint after completing a level. Temporary checkpoint mid-way through during the current game.
FTL: Advanced Edition
There is some reading in FTL, but you can skim it if you aren't interested.

You kind of perfectly described Tower of Guns.

That's my recommendation, too. No save feature (unless an update added that), but it doesn't need one.