The best English suits, from Savile Row, have sleeve buttons that fasten with button holes. Why ? (Cheaper suits have this but the holes are 'dummies')
Think of cufflinks ( cuff buttons). Were they introduced to stop men/boys wiping their noses?Er, no.

Can you get a hand through the sleeve with them fastened ? No; you have to fasten them later.It's the same for ordinary shirts usually.
Imagine that coat ('jacket') sleeves were once narrower than 21st century ones ? If they were how would you have got the coat/ jacket on ? Correct

By the late C18 sleeves on men's coats were narrow. They were fastened over shirt cuffs,usually lacy, which spilled out; it may well be that buttons were needed before too but here we may have a line of buttons, like a lady's long evening glove.
Does it sound likely that the Royal Navy would redesign its uniform to accommodate snotty boys? Or do you think they'd have been either ignored or thrashed for ill-manners? And what of others, guards say, the immaculate soldiers with 'gentlemen' for officers. Hardly in need of nose wiping insrtruction. (But it wouldn't work anyway; a shirt sleeve has a button !)
Gentlemen carried their handkerchief in their sleeve.( Nelson's boys could have done so, too) My grandfather's generation always did; some of mine do; a few young fogeys still do. There are no pockets for a usable handkerchief in a morning coat ( or any tail coat), the business attire of their youth, which may explain this; though the old reason be lost now. There was not even a top pocket in a formal uniform coat. (When seated pockets in trousers can be pretty inaccessible, too).
So that's the real reason; the others are funnier though!

Incidentally,the older generation and traditional tailors here still call the 'jacket' 'the coat'; this usage seems to have been current in the US in the 1930s still ( 'A coat and two pairs of pants' on an economical suit; Oliver Hardy saying that Laurel had ruined his 'coat' from his suit and so on, come to mind)My great- uncle wouldn't recognise 'jacket' and would chide that "only potatoes have jackets" (' a jacket potato' is one baked in its skin).
