Is your objective to have a working band, or to be an instructional group teaching people/yourselves how to perform? If the former, you need a different person.A natural sense of timing is something not everyone is born with. I believe that if one works at it that improvement is possible. But if this singer is already trying to be in a working band it seems to me he's skipped one of the basics. ...
I have found that singing pickup notes seems to be one of the very most difficult things for amateur singers to do. I don't know why; I'd guess something like half of all the lines of lyrics ever composed and sung have pickup notes.
We're dealing with the same population here that will insist on clapping on 1 and 3 on backbeat songs (and always just slightly off time).
The way most song sheets are written doesn't help either, because the words for the pickup notes are generally typed at the beginning of the line, when for clarity they really ought to be at the end of the preceding line. When I type song sheets I always boldface the word (or syllable) that's on the ONE, and when I use someone else's typed sheet I often underline or circle the ONE. (I'm not a very good singer and I need all the help I can get.)
I think that as a working band you have the right - nay, the duty - to demand that each performer meets certain base level standards of ability and skill. A singer who can't come in on the right beat does not, in my opinion, meet those criteria.
Now if what you've actually got is a social hang with instruments, or if you've set up a joint learning workshop where the main purpose is for you all to learn how to do this stuff better, THEN you can start talking about how to teach the singer how to sing. It's not just tone quality and remembering the words, it's RHYTHM too.