I'm not sure that people are saying (or not always, anyway)-
'chester drawers'
I think people are possibly saying 'chest o' drawers' - which is archaic, but not grammatically incorrect. Contracting 'of' to o' isn't used in print anymore, just like contracting 'over' to o'er isn't, but they were both shortened like that until recently- you can sea it older pieces of poetry also still in 'Portuguese man o' war', etc.
We don't realize that's what's being said- because that contraction is archaic in written english- but I think spoken english tends to retain archaic forms longer than written english- for example- in some parts of Yorkshire, people still use 'thee' and 'thy' ,(instead of 'you' and 'your')- these are dialect differences -but they are also perfectly grammatically correct but just generally considered archaic, in a lot of places, and in everyday written english and print.
I think this is actually what the alleged and supposedly wrong , 'chester drawers' is- it's actually 'chest o' drawers' which is old fashioned, but not incorrect, and that if people write it 'chester drawers' , they're just making a spelling mistake- by writing it phonetically, because they don't quite realize what they are actually saying.
So, basically, I think it might be ok, tbh.