I have a 2019 MBP 16" that I received from my University in Nov. 2021
The unit arrived to me brand new (never opened) and never configured. My university uses software delivered online and authenticated through an iLok USB dongle, so there was no need to pre-load any software. It initially booted into Big Sur, but by that time (Nov. 2021) I was immediately asked to upgrade to Monterey and since I hadn't loaded any software yet, I did exactly that on Day 1 of use. A few days to no more than a week later, the university's technical support began to caution students NOT to upgrade, since a vital software for many courses was incompatible with Monterey.
I delayed downgrading, thinking, I just started classes and the developer will have 8 months to fix compatibility.
So, I had to scramble while finishing projects in another software to downgrade the OS over a weekend, so that come Monday with the start of my new class, I'd be able to install the required software and keep up with the class. I should mention that I'm an online student with no alternative except my 10 year old PC - which simply won't perform adequately. 
I didn't have the option to use an External Backup, but I had plenty of free space on the Internal drive, so I made a partition and put my backup there. I then searched Apple Support and only found BigSur links to the AppStore, which would download as a .dmg file. My first attempt seemed to follow what Apple support suggested, "after backup, launch into recovery mode, erase the Monterey partition, and use Internet recovery to reinstall BigSur". However, Internet Recovery didn't have BigSur, it only had Monterey, which is puzzling since the MBP was originally BigSur. Nevertheless, I had no choice but to reinstall Monterey, load my backup, and try this process differently.
My second attempt used 3rd party guides (with similar steps among many different host sites) to add a terminal command, which was meant to isolate the BigSur.dmg and make it bootable. So, after I go to recovery mode I still would erase the Monterey partition and keep my backup partition along with the BigSur bootable partition. An extra step at this point was to change the setting on my Startup Security Utility that disallowed booting from external media. The guides pointed to the terminal command that made the .dmg bootable which necessitated this setting change and that the MBP would only ever find Monterey in recovery mode otherwise.
I also decided to enable my Firmware Password at this time since it was my understanding that if the boot mode setting was incorrect, the firmware password would allow me into the Utility to correct it, or at the very least allow me to reinstall Monterey if nothing else, like my first attempt. It wasn't the span of 2 minutes between when I confirmed the password (a common password I use often) and when I was faced with the Padlock screen. However, THE password that I set failed to work. I'm not very familiar with the behavior of this screen, but after I would type in the password (displayed as hidden dots) and press Enter/Return, the password field would just disappear. It was as if my Enter key was a 'Clear' key instead and the MBP would offer no response.
I tried alternating case for the password, assuming I'd made a typo or engaged the CAPS Lock instead of Shift, but every iteration of THE password that I tried would fail. I tried using the mousepad and my external mouse to click the submit arrow instead of using the enter key and still my MBP remains locked after exhausting the likely possibilities (common typos included). And, yes, I tried to submit a blank or null password, tried my Apple ID password, tried my Mac Account password, when I say I exhausted every likely password available, I feel my attempts went into absurdity. 
I understand how simple and dismissive it can be to forget a password. I've forgotten plenty in my time and I've worked as a support agent to help many others reset theirs. However, this issue can only at this time be classified as "Not Working As Intended" because if I type a password twice to satisfy the confirmation, I should have 100% confidence that same password will grant me access when it's being utilized.
My nearest facility is not an Apple store, but an Apple partner, and at first they directed me to the Apple store which is +20 miles out from their location, but for whatever reason, the tech I called prior to making the trip decided he would attempt the repair. I forwarded the proof of ownership email that I received from my university to the tech's store email address. If there was any question about ownership, it should have been clear coming from a pdf generated with the university masthead and being sent from a .edu email address that matched my legal name. However, quite abruptly the tech asked me for a fee and I assumed he didn't have a system to denote Covered products, but this MBP was under AppleCare+ 3 Year Protection and was still well within the 1 year warranty. I refused to pay the $149 fee because I simply couldn't afford it and to be surprised with a fee after driving over an hour was intolerable. 
I can understand if this issue is treated the same regardless of why the password doesn't work, but my question ultimately is: Why is there a fee associated?
If you made it this far, take a deep breath...I've had to suspend my classes to a time that my budget can accommodate for the expense, so as much as I'd like to offer a succinct issue, doing so in this situation would massively underplay the exact issue. Perhaps this glitch is 1 in a million and we may never see this happen again...All the more reason not to soak the consumer in exorbitant fees and if anything, else Apple will you please put up some better guides for OS downgrade procedures?