I’m thinking something along the lines of the old Star Wars video game where there is an X you move with your mouse and you just shoot at targets. Something that is quick and fun to play for a few minutes at a time. Anything like that?
If you start up Chrome without a network connection, the “no network found” screen has a running/jumping game. Hit the space bar and the T-Rex will start to run and you have to jump over rocks and cacti.
There are tons of Atari, Sega, and even arcade machine emulators that are available in browser online. Many classic arcade games and home video games. For the arcade games, there’s even a requirement to put in a quarter (press a certain key so the emulator thinks you put in a quarter). Also, old Apple ][ and Amiga games.
I can’t search for these right now, but it’s really easy to Google.
Classicreload.com is but one of many.
These are running slow on my Chromebook, but maybe you’ll have better luck on a less underpowered computer:
Atari redid some of their classics as flash versions. Missile Command or Asteroids might meet your needs:
Digging through that site was what actually inspired me to post. I was looking for some specifics people have played if not there then elsewhere.
Here’s one. Aim with mouse and move left/right with cursor keys (or A and D keys).
It requires Flash.
Have you tried this one - Onslaught-2? It’s my favourite tower defence game. To play it you’d just need a Flash Player. 
Not shoot, but there’s Pedestrian Killer (don’t use Flash, so I can’t vouch that it’s the one that I mean or that it works).