I found the data for admissions for PHE (sorry, couldn't find the other countries) and the average would appear to be slightly lower than 204,000, but I agree that Covid patients have only ever been a relatively small proportion.
Problems have been caused because there have been Covid-related staff shortages, patients have needed intensive nursing and beds have needed to be socially distanced, so capacity has reduced. Not only that, but patients usually stay in hospital for longer with Covid than they do for many other conditions.
1,000 extra patients a week or (heaven forbid) 1,000 extra a day would put considerable pressure on the NHS because it needs to catch up with elective and "non-life threatening" procedures. Even with improvements in treatment, it would be likely that deaths would be in three figures and that's before the effects of Long Covid are tackled.
The biggest problem is that the NHS was already under-resourced and on a knife edge.