Those get picked up in the SE taxes.
I am also not saying that 1099-NEC is the best-practices way to go, but to my mind it's more important to get the client straight going forward if that's the way to get them to agree. Plenty of people will not agree to do wages, file now-late 941's, and all the rest - but they will agree to a 1099-NEC plus PR starting Jan 1 of the year (already back-dating start of PR to the then-current year, since these people tend to show up in March). I would explain - in writing - the possible consequences.
In our experience, the no-PR problem has mainly been new-and-small companies who are not making a ton of money (if even running at a profit at all). Straighten them up as best and as quickly as you can, and have a loyal client for years to come. Remember, we can only advise on changes prior to meeting the client; we cannot compel. Draw your own lines that you will not cross, but as long as the client fixes going forward, I would most likely agree to 1099-NEC the owner(s) if they balk at doing prior-year payroll with late 941s. Because we know most of them would then take their papers to some shyster who will let them continue to mess up for years to come. Better to get them in current compliance and keep them there.