Your suggestion would be a database? Can’t see myself paying for a big database just for that, perhaps Trello can be used as a logging service.
Trello’s intent is only for basic kanbanning and personal planning. Trello would be an even worse option for logging and databases because it’s not designed to support those use cases in any sense of the words and it does not scale like a database does, be it speed or anything else.
Trello has a help article on troubleshooting slow boards which advises having fewer than 1000 cards assuming you’re only using the bare minimum features of a card. If you’re planning to work on an experience at scale supporting moderation features like this, you do not want to deal with how slow Trello can get, let alone the lack of features it has to support database use cases.
Additionally, from what I last recall, Trello silently implemented a special rate limit against the Roblox UserAgent which imposes a 90% lower allowed calling limit per interval. This information I know to be to date as of December 2020 when I was working as the lead developer for a 250K+ member group which abused Trello for a lot of its web features (applications, user-written library, special ranks, etc).
Like I said in my post, my suggestion would be to search for proper services to serve your use cases and see if the free plans suit your needs or if you can build and host your own database. There are a lot of services with free plans that you can fit into, or if you need storage then it’s fairly cheap. For example, see Cloud Firestore pricing and look at the free quota as well. It could be possible not to exhaust the quota for a small experience provided your Lua-side does some heavy lifting so as not to query frequently and you’ll probably be netting enough to pay for additional uses for a large experience with major traffic and revenue flow.
Then my best bet would not to have logs outside of the game.