Now that citizen stats make a difference, and we’ll soon be having citizen traits as well, I think that when you get a new Hearthling to say yes or No to, you get to see their stats and traits. I don’t want to be letting “Over-eaters” into my Town! 
Also, after reading all the replies, I think the new Hearthlings should already have a class (Like in good old Alpha 4), with specialized skills in that department, even though their profession is still changable.
How dare you, sir! 
It would be useful, though, adding an element of ‘Do I accept this person now, or can I afford to wait a bit longer?’
Choosing who you do and don’t want in sounds interesting! Settlers could have bad traits only, a mix of good and bad traits, or, on rare occasion, good traits only! This could have an affect on class selection in two ways. Necessary and preferred. A preferred trait is something that is never necessary for a class but always nice to have. For example, a “genius” or something would be good for an engineer. Necessary would be mainly for magic classes. For example, only people with the trait “magic blood” could be a magic class. Also, certain traits could be linked to each other. For example, "that guy is an overeater, but he also has gigantism, so he would be a good addition!
I think its also worth considering a penalty for refusing someone… maybe a ‘gap day’ where nobody offers to join?
Well, they still have to wait another day for another chance at it. I think if it is balanced correctly then it will be a weigh up of how picky you are vs how much you want another Hearthling. In effect it will be the same penalty you receive currently for not meeting the criteria of the daily update.
It could also depend on how often you deny entry to your town and the stats of your current settlers. If you have a majority of near-perfect settlers stat wise and you keep denying settlers who have mediocre or bad stats, the daily updates may require much higher food and money for you to be able to accept or deny new members.
Timber and Stone has a system like this where you can look at settler traits before you accept them. One is actually “Over-eater”, and apparently people say never to accept over-eaters
in t&s i have lots of over-eaters (ok i have also lots of food xD) - there are other traits that are more problematic xD
And what about :
Thanks everyone for the new replies! Look at original post for another idea I had.