Based on this answer from Grace Note to a related question, seems that the main reason behind all migration target expansion requests begin rejected in the past was "we don't want to clutter the popup".
We don't ever want to show more than 5 here, because it gets overwhelming.
Thus, in order to get [a different site] on there, you'd have to make a case that we can kick one of the existing 5 off the list. Are you prepared to do that?
Currently the "accepted action" when a site isn't available in the "Magic 5" list is to flag the question with a custom text flag and leave a notice for the desired target. Could be enough for most case, except that the "flag category" is not so clear (NOTE: I don't have the req. rep to see the flag review page, so I don't know if this makes much difference).
Your proposal would solve the problem of clutter. A single autocomplete field, like the one on the data dump query site, would be enough to target any site forever and ever, and won't add much noise to the popup.
That said, as other user explained to me in comments and such, it would seems that the problem today switched from "too much clutter" to "too many wrong migrate flags". If that is true, your solution won't do any good to the system: having a field would only allow "arbitrary" migration to increase, thus increasing the number of wrong migration.
It would seem that currently we are actively trying to discourage migration. If that is the point, any request that would make migration request easier is bound to be rejected.
I still think that in some case your proposal could help. Maybe we could insert some high rep requirement? Don't know. We need some staff member to elaborate on the current situation.