Have these services actually turned you down? Usually this is a marketing thing where they prefer work email on forms to assess commercial opportunity, but I’d be surprised if they refuse a personal email.

Why would you need specifically "cloud" logging for that? Spinning up grafana and loki is rather trivial in the modern containerized world and that'd cover 90% of what you want from logs. Neither is a resource hog, too, it's so much better that e.g. the ELK stack for logs that you only look through occasionally.

Have you looked into grafana cloud? I haven’t used it myself but they have a free tier. Or if you’re open to self-hosting you can run loki and grafana yourself.

Does render fit the bill? I'm using it to run a mastodon bot using Node and in free instances it stores logs for 7 days iirc.