1) A preliminary round is not a patchwork solution. It is standard practice in almost any cup competition to use a preliminary round to reduce the field, rather than arbitrarily excluding teams.
2) It is a long-term solution. It is quite unlikely that BB will go away from its 1:4 or 1:2 pyramid, which means that the total number of teams will not be a power of 2.
The field is not excluding teams arbitrarily. It is excluding the weakest teams, which believe it or not is also standard practice in almost every Cup competition.
Moreover, it is a patchwork solution, because as the number of teams in the championship increases, the percentage of excluded teams, as a percentage, is decreased. This fraction closes down to about 25% pretty quickly. Therefore, you are suggesting a fix that you find necessary for 256- or 1024-team tournaments, and this is why I call it 'patchwork'.
I don't think a preliminary round will add much to 4k and 16k team tournaments. As a matter of fact, it will be a wasted week when countries expand and the Tournament takes closer to a full season than to a half.
I've mentioned before that I am fine with tournament participation not being available to every single team. As a matter of fact, I don't think you can have a reasonable pyramid where your teams will be an exact power of 2, so...
"I don't know half of you half as well as I should like; and I like less than half of you half as well as you deserve."