There is a local chess school here that offers a league tournament, which is very popular. It's also one game a week in one night. I have not heard many that grow out of that league.
Playing a game is better than playing no game, but it also gives wrong impression to the parent, that "my child is playing tournament every week". With this impression, the parent will not actively search for another better tournament. They will go to that club week by week, be satisfied with the activity, but not the result. If you are a retiree, this is not bad. But if you are a fast-growing youngster, this is not good.
Some TDs report results week by week, while other TDs report results by combining multiple weeks. But you only get one practice chance every week.
Doing this doesn't save you any time, because you have to drive to the tournament every week. That night may be all for chess. Your child will have to make up his/her homework on other days. The total time might be much more than a tournament on weekends.
The big disadvantage of playing one game a week is that many players need time to warm up. They need one or two games to get into chess playing mode. Every time they get into mode, then they stop and go home. They have to re-start next week. But same thing might happen next week. We all know, chess is like war. It does not only depend on knowledge and skill, but also on morale and mood. If you have confidence, you could easily outperform yourself. For a kid, building that confidence is critical. This requires several games in one tournament in a short time span.
Go find multi-games tournaments on weekends. Do not split them out.