Copenhagen layover

I am travelling via Copenhagen to Budapest on a single ticket and have a 10 Hr overnight layover in Copenhagen.

I would like to leave the airport to get a hotel for the night but am unclear on whether this is possible. I am a UK born, UK passport holder and our Government website states entry to Denmark is visa free.

Having spoken with the airline, airport and Danish UK embassy I am still unclear. The airline and airport deferred to each other/embassy, whilst the embassy said I need a Schengen visa and when I raised that their website states British born citizens do not need a visa they have not replied.

Does anyone have experience here? Particularly with the new entry rules. TIA.