This is great! Hopefully the lambda you're using to aggregate your error logs doesn't have any issues. Though lack of signal is a signal, too.
We use logz.io which is a managed elk stack. It has a free tier with 1 day of log retention. They have a lambda log shipper that sends new cloudwatch logs to your dashboard where you can filter, search, and setup alerts.
Oh I get that for sure. My real recommendation was for the elk-stack in general. One can self-host it pretty easily depending skillsets. The service I recommended just makes it about as easy as your email aggregator.
The email aggregator is great but if you need an actual way to search through your logs, maybe after the email aggregator alerts you that "some requests failed" doing it in cloudwatch is not so fun.