Tim, you are right - this setting they have is a bit unfortunate for such a great treasure. However, in practice searching for the problem's name in GoogleScholar works fine (use advanced search to restrict your search to the journal, etc.), so I am not sure there is a great need to compile the master index. BTW, other journals have also many interesting problems, notably SIAM Review (I especially like Problem 66-11 which now has its own Wikipedia page).
About problems with/without solutions. There is quite a bit of neglect here. As I explain here, Dyson published his famous "partition rank conjecture" as a Monthly problem, and that problem motivated Fine to prove his famous identities. The problem was solved by Atkin and Swinnerton-Dyer about 10 years later, but you will never hear of this on the Monthly pages, I believe.
In another interesting example, Problem 4325 is a classical "inscribed square" problem (see here and here). Even though it has a "solution", it is important not to take it too seriously.
P.S. One bonus comment: there is a Russian book of Monthly problems, from 1918 to 1950, all with solutions copied and translated from English. If your problem is this old, you can look up the date of the solution in there (and then read it in English).
UPDATE Another example just occurred to me: in this nice paper the authors explain how Erdős posed Problem 3763 (on convexifying polygons), how Sz.-Nagy's solution was incorrect, and how a generation of researchers tried (and, occasionally, succeeded) fixing this solution. The point is that it's basically impossible to do this kind of followup on every Monthly problem.