Would British citizens need to get their passport checked at the Ryanair visa desk from Jan 1st 2021?

This should have been a comment, but is suitable as an answer. Actually, it is a guess and is not based on reliable information.

Airlines, as we know already, want to check visas thoroughly to avoid the cost of bringing refused passengers back.

Airlines are also accountable for the usage of the airport facilities (gates) and the delays in flight. Rule of thumb: airlines pay more money to the airport if the flight gets delayed.

The back-end fare system is a lot complicated because airlines can choose or not to use their own ground staff or hire local staff. Indeed, low-cost companies like Ryanair succeed in reducing the costs to get their beloved revenues. This includes optimizing the utilization of human staff.

It is then intuitive that the documents checks at the check-in desk is likely way faster/cheaper than the check at the gate, in particular when compared to the risk of discussion with the passenger and consequent delays. Remember: a passenger with invalid visa requirements will attempt to their best to convince the attendant to let them go.

Of course airport security staff won't prohibit you to enter the secure zone and approach the gate if your boarding ticket was not stamped.

I don't know what happens if you skip visa check, but anything may happen from harsh lecture to direct no-questions-allowed refusal.

For entry into the Schengen Area (not within the Common Travel Area), British citizens will be treated as third-country nationals.

A entry/exit stamp will be placed in their passport to enforce the 90/180 days rule.

An exception will be made for those presently (2020) inside the Schengen Area. The 90/180 day clock will start on the 2021-01-01, so the lack of an entry stamp will not automatically assume an overstay.

Should, for any reason, no stamp upon entry be given it is advised to retain some proof of date of entry (ticket etc.) to avoid complications when leaving.

The same is true for (non-Irish) EU Citizens when entering the United Kingdom, especially when arriving from the Republic of Ireland. At present it is not clear how the date of entry will be noted for those travelling with only an ID (which will still be possible until 2021-10-01), afterwhich a passport is required for non-resident EU Citizens who are not Irish.

As to how private corporations, who will be held responsible for transporting persons without the proper paperwork, will deal with this is their affair. Assume they will refuse to allow you to board if you don't fulfill their conditions.