That risk is the same for almost all teams because there are all those tournaments that inevitable bring a lot of games. Whether it is Conference League. Europa League or the CL, everyone has at least those games from the group stage or league stage and then the round of 16 games. The only problem I see is if teams get unlucky and have several injuries without impact from their opponent players. That would be a sign of fatigue, but if they get injured because of attacks in game, there isn't much you can do about it.
That's why squad depth is quite important because the coach can make a number of rotations at certain times of the game. Almost all teams will experience conditions like this but how a coach can see the opportunities to solve them. Almost all teams that play in the Champions League participate in a number of other tournaments as you said and injuries to several key players can cause quite a big problem. Like what happened to Real Madrid in the previous season and they failed to achieve the best results in winning the title because they were affected by injuries to several key players.
It also depends on which player gets injured too. Like if you are Real Madrid, just 2 injuries, for one week, could end your season completely. How? Well if Vini and Mbappe gets injured, right around at last 16 games or quarter final games, then we are going to see them have some trouble and fail. So just mere 2 injuries, will make them lose, even if they have bench, none will be as good as them.
This is how you lose, bench and depth do matter a lot but injuries could mean a lot. Real had a lot of injuries this season that made them lose so many games, hence why they had a bad year. It's good to add some depth, like in case of fatigue and shape of players and rest some of them, but on important games, you need everyone to be ready.