I think the word you might be looking for is "coincidental" (possibly "contrived").
And perhaps in contriving this remarkable coincidence, Tolkien was making his first essay into the concept that there is no true coincidence in Middle-earth, that all things happen for a reason, even if that reason is never known. Frodo coming of age at this particular time -- when Bilbo was finally feeling the negative effects of keeping the Ring -- made it easier for Bilbo to pass it on to Frodo, along with Bag End and his other possessions. Frodo needed to be old enough to be totally undisputed as Bilbo's heir, and Bilbo needed to be old enough for other hobbits not to question the fact that he never returned (especially since he had gone off once before, and returned at a most inopportune moment for those who had been buying his possessions). Contrived, certainly, but one wonders whose finger within the subcreation caused it to happen in just that convenient, coincidental way.