> Does it automatically filter my email into tabs for primary, promotions, social, and updates?

This is where email MUAs[0] shine. Mail user agents such as Thunderbird[1], KMail[2], Apple Mail[3], and nmh[4] (for hard-core Unix command-line aficionados) support filtering and automatic categorization to varying degrees.

All while being mail service agnostic.

0 - https://en.wikipedia.org/wiki/Email_client

1 - https://en.wikipedia.org/wiki/Mozilla_Thunderbird

2 - https://apps.kde.org/kmail2/

3 - https://support.apple.com/mail

4 - https://www.nongnu.org/nmh/