You could go to lichess.org and create your own time controls. They let you stipulate an initial time and an increment.

For example you could have a game with a 30 second time for each player and then add 15 seconds for each move after that. Increment ranges go from 0 to 180 seconds.

15 seconds plus 5 seconds per move should be doable.

I believe the time would be banked for each move rather than a hard fixed limit for each move.

You could then create a custom game open to others, or make the game available by link to a friend.

The closest match that comes to mind would let your players bank the time they don’t use. Using an n+5 configuration. The software cutechess may be capable of it but it’s not online.

But what bugs me is it would be a goofy way to play as different positions can demand more thought than others.

You would probably need to make your own custom rule set and host it yourself or do a regular game on something like lichess and enforce the time constraints outside of the platform.