Can I filter clients by whether they've sent their records, and bulk-email the rest?

Not directly. There's no "records received" filter on the client list, so you can't filter by records received and bulk-email the rest in one go.

Job status filtering, including grouping by status on the Kanban view, is only available from the Jobs screen. Use Jobs > Filters > Status: and choose Records received (or another status, such as Awaiting records). The Jobs screen has no bulk email, so this can't be combined with bulk client email from there. This has been logged as a feature request. To request it yourself, use Need help? > Tell us what to build next > Suggest ideas.

Bulk email is on the client list: select the clients, then Clients > message > Compose e-mail.

You can still bulk-activate a service for a group of clients using Clients > bulk actions > Edit client or service data.