Install later version of Apple Mail 9.3 on El Capitan

No. Only if your iMac can be upgraded to macOS High Sierra (or later) and even that operating system is getting too old to reliably connect with Gmail servers owing to deprecated security protocols. Your plan B would be to access your email via a Firefox Extended Release (ESR) browser.

Not sure any of this will help, but it might, is your gMail MAP?

Likely you won't need all of this info...

Mail and Google's OAuth 2.0 policy

Mail and Google's OAuth 2.0 policy - Apple Community

Fix…

https://code2care.org/macos/fix-mail-authorization-error-400-invalid-request/

Mac's Mail App is not as modern as iOS's Mail app, Many eMail providers now require a separate App Password different from your eMail password…

If it's an IMAP account, delete it in Sys Prefs Internet Accounts then use "Other" instead of gMail choice to add it.

Most likely biggest factor is using Other in Mail setup, then using a Browser that doesn’t have gMail PW stored init to get the App PW.

Unable to add Gmail to MacOS mail - Apple Community

Thanks to Bob Sabiston…

iMac says Connection Insecure. Failed to … - Apple Community

Thanks to Webmb…

Sonoma can’t connect to gMail…

I'm no longer receiving mails on my iMac … - Apple Community

GC mac

User level:

Level 1

4 points

Nov 1, 2023 10:12 PM in response to Maggiellv

I also spent hours trying to solve this issue until I followed this video... It worked and my Gmail accounts are back on Mac Mail.

Give it a shot and good luck! https://www.youtube.com/watch?v=U0WPw2TsTAY

It might just need an App Password separate from the gMail password…

The process to add a gmail account to my … - Apple Community

https://support.google.com/a/answer/9003945?hl=en#apple-mail&zippy=%2Cset-up-gmail-with-mail-on-a-mac

https://discussions.apple.com/thread/254920541?answerId=259158920022#259158920022

https://support.google.com/mail/answer/185833?hl=en-GB

Apple Mail won't sign in to Google account - Apple Community

my iMac has OS Yosemite 10.10.5 Does this… - Apple Community

my iMac has OS Yosemite 10.10.5 Does this… - Apple Community

gmail on mail Version 11.5 (3445.9.1) - Apple Community

gmail on mail Version 11.5 (3445.9.1) - Apple Community

Note from PrestoWebDesign…

https://discussions.apple.com/thread/253701684?answerId=257151754022#257151754022

Trying to add Gmail account to Apple Mail… - Apple Community

https://blog.google/technology/safety-security/the-beginning-of-the-end-of-the-password/

You sign into your Email provider's site (e.g. gmail.google.com) and access your email through the browser, providing it meets the security requirements of your Email provider.