Why would my mail suddenly start failing due to "cannot connect to server?"

If you've already wiped the machine, it will likely clear out any lingering faults or glitches but battery drain can be caused by several things, not least of which a rogue application or a bad battery.

Have a look at the energy consumption of your Mac if/when you see the issue happening, it will tell you which app(s) is/are pulling more heavily than others. You can check the health of your battery in macOS pretty easily, it will tell you if there is a possible hardware fault with the battery.

From what you've described so far, it sounds like one of the issues may have been Spotlight was stuck in a loop and the index needed to be rebuilt; it's not common but can happen once in a while.

If you continue to see the issue(s), please feel free to post back.

Welcome.

If your Internet and/or Wi-Fi access is dodgy, that would cause your e-mail access to become flaky.

After you've ruled out Internet and/or Wi-Fi access access issues, if your e-mail issue is specific to one (1) e-mail account only (e.g., iCloud vs. others you may have like Gmail, Outlook, Exchange, etc.), start by restarting your computer, then visit this page: Get help using iCloud Mail

If you have a lot of e-mail, especially if you have multiple e-mail accounts turned on (e.g., iCloud, Gmail, Exchange, etc.), it's likely still downloading, organizing, and/or indexing the e-mails.

Check in your Mail app if it is still downloading by opening the Activity window. If so, that's very likely why your Mail app is chugging along so it's best to let it finish while on AC power until it's finished.

Meanwhile, if you have Spotlight indexing (esp. after a fresh install of macOS), which is normal if you've just done a clean install and restored your old data to the machine, this can take some time, depending on the amount of information being indexed.

If you look in the Activity Monitor, turn on All Processes, you're very likely going to see one or more instances of "mdworker" and/or "mdworker_shared" running in the background; it's likely because of the recent install and/or Mail app downloading and rebuilding the mailboxes.

Both of the above scenarios will have a deleterious effect on your battery life, especially Mail if it's still downloading and rebuilding the mailboxes. Mail will usually wait until you're on AC power to finish downloading large mailboxes but if you're running anything on Microsoft Exchange (e.g., Work E-mail), it may ignore this request.

Meanwhile, you can always run Apple Diagnostics to check for hardware faults, including the battery. You can also check the Battery Health in macOS.

MTLCompilerService has to do with the graphics architecture/video; it likely spiked when you opened a new window but it's normal to have several instances of it running.

You can always turn the refresh interval up to 1sec to get a more accurate yet aggressive accounting of everything.

As for Mail, you can let it continue and keep your mail, it'll just take it a little longer to finish the larger the mailbox. The Genius Bar can look over your machine for anything else outstanding.

Meanwhile, it may be worth a shot to reset the SMC as it regulates the power-related functions of the Mac.