MacBook Air M1 startup options not working

There are a few things in the descriptions provided that seem very off and not right.

In- light of this >>

Suggest downloading the Application Etrecheck directly from a well Respected ASC Contributor. The application is free or paid from added features. Run the application with Full Disc Access ( Security & Privacy - Full Disc Access ). It will take a Snap Shot -  both the hardware and software. The Report will Not Reveal Any Personal Information. Post back the Full Report - copy and paste - using the Additional Text Icon ( 3rd Icon to last )

We can have a look at the report for possible issues and may have possible suggestions to resolve the issues.

Any Third Party Applications that will interfere with the normal operation of the OS, alter, modify, remove or delete or attempt to do so is an invitation for disaster and may require a Reinstallation of the OS.

At this point I'd Boot into the Recovery volume (boot with the Command + R keys held down) and reinstall the system. As with any big install make sure you have a full and recent backup of your data files before hand. Usually the reinstall from the Recovery volume goes without incident. I've done it several times without issue.

Point # 2 - Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal operation of the OS and may even Corrupt the OS requiring a Reinstallation. The Built-in Security of Big Sur is all that is required.

There are no known Viruses in the wild that self replicate and affect macOS. There are Malware and Adware that does affect macOS and are often times downloaded as part of an Application from Third Party UnTrusted Site and get installed along with the Application .Suggest downloading from a Respected ASC Contributor the application Malwarebytes for Mac. It is free or paid for added features. Run the Application and it should remove the malware / adware.

Evertyhing in List Below should;d be removed as per Developers Specific Instructions

/Library/LaunchAgents/com.mackeeper.MacKeeper-Reminder.plist

        Executable: /Applications/MacKeeper.app/Contents/Library/LaunchAgents/MacKeeperAgent.app/Contents/Library/LaunchAgents/MacKeeper Reminder.app/Contents/MacOS/MacKeeper Reminder

    Launchd: ~/Library/LaunchAgents/com.mackeeper.MacKeeper-Info.plist

        Executable: /Applications/MacKeeper.app/Contents/Library/LaunchAgents/MacKeeperAgent.app/Contents/Library/LaunchAgents/MacKeeper Info.app/Contents/MacOS/MacKeeper Info

Other] com.mackeeper.AntiVirus.plist (? c4e6864e - installed 2021-08-23)

    [Loaded] com.mackeeper.MacKeeperPrivilegedHelper.plist (KROMTECH ALLIANCE CORP. - installed 2021-08-23)

[Loaded] com.mackeeper.MacKeeper-Info.plist (? 0 - installed 2021-08-23)

    [Loaded] com.mackeeper.MacKeeper-Reminder.plist (? 0 - installed 2021-08-23)

    [Other] com.mackeeper.MacKeeperAgent.plist (? 0 - installed 2021-08-23)

2021-08-20 CleanMyMac X (4.8.6)

First, there is no reason to ever install or run any 3rd party "cleaning", "optimizing", "speed-up", anti-virus, VPN or security apps on your Mac.  This user tip describes what you need to know and do in order to protect your Mac: Effective defenses against malware and other threats - Apple Community.  

There are no known viruses, i.e. self propagating, for Macs.  There are, however, adware and malware which require the user to install although unwittingly most of the time thru sneaky links, etc.   

Anti Virus developers try to group all types as viruses into their ad campaigns of fear.  They do a poor job of the detecting and isolating the adware and malware.  Since there are no viruses these apps use up a lot of system resources searching for what is non-existent and adversely affect system and app performance.

There is one app, Malwarebytes, which was developed by a long time contributor to these forums and a highly respected member of the computer security community, that is desshoigned solely to seek out adware and known malware and remove it.  The free version is more than adequate for most users.  

Also, unless you're using a true VPN tunnel, such as between you and your employer's or bank's servers, they are useless from a privacy standpoint: Public VPN's are anything but private.

That being said you've installed two of the most egregious of those apps, MacKeeper and CleanMyMac:

 Uninstall then according to the developer's instructions. When done you can check to see if you've removed all of the supporting files by downloading and running the shareware app Find Any File to search for any files with the application's or the developer's name in the file name.  For mentioned software you'd do the following search(es): 

1 - Name contains mackeeper

2 - Name contains cleanmymac

3 - Name contains macpaw

Any files that are found can be dragged from the search results window to the Desktop or Trash bin in the Dock for deletion.

FAF can search areas that Spotlight can't like invisible folders, system folders and packages.  

Then run another Report and let us see.

As of the Reported Supplied and therefore advise offer was based on this Report - that which you think was removed are still Installed and lingering on the computer.

You are suppling the Same Report ? First Report EtreCheckPro version: 6.5.1 (6F003)Report generated: 2021-08-31 21:42:30

2ND Report EtreCheckPro version: 6.5.1 (6F003)Report generated: 2021-08-31 21:42:30

The problems being experienced now, may or may not be the after affects caused by the MacKeeper and CleanMyMac software that was installed. The level of damage by either or both of these two applications can be very extensive.

At this point - if I were you and I am not - would wait for @ Old Toad to jump back in with further advise and / or directions.

Personally - would take a very hard line and do a drastic move the remedy this situation. But wait and see what Old Toad has to say.

Use macOS Recovery on a Mac with Apple silicon

Why are you trying to get into Recovery Mode ?? Is there something specific to be achieved ??

One does not require getting into Recovery Mode to restore personal data accidentally removed.

A Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS. 

You do not have to do a complete system restore just to recover a few files:

Recover files using Time Machine and Spotlight on Mac - Apple Support