I think the solution is a bit Draconian in that it would unfairly penalized legitimate injuries. For example, if a kid gets a cramp - which is actually very common during hot games - I think it would be unfair for that player to have to sit out the entire series.
Why unfair? Might help with pregame hydration. In fact due to the tendencies for repeats of that issue, we might need a cramp protocol developed. For sure a second injury timeout for a player should be a game ender.
Possible pieces to consider mixing and matching. Something easy to administer.
First one each quarter is free.
One for one sub only during any stoppage. Team with injury must remain in huddle between the numbers
Second and > each quarter is 5 minutes for each player, Ref can put a sticker on the helmet on the field with return time. Tape is cheap. Can only reenter at a dead ball when a ref can be approached and remove the sticker.
Second in a game for a player take his helmet, done.
2nd and beyond in a series is a team time out on top.
When a TO is called for but not available, the other team gets choice of 5 yard penalty or a TO added back to them.
On any down but first, an automatic first down. Just move the chains and flip the marker.
I bet a lot of those guys with cramps and stingers would hop off rather than lay down.
Anything you do runs the risk of hurting honest injury occurrences, but it is what it is.