Just for comparison's sake, if there's an England national team caliber player in the 3rd division, don't you think they'd end up in the EPL pretty quick?
I'm not so certain I follow you, Brian.
Assuming there's an English team in division III that has a player of national team caliber (or even is on England's NT), you seem to contend that said team would be a shoe-in for advancement. Well, at present, this is likely the case, as England has but a handful of active managers, and likely has a rather limited player pool (I can't really say I know; tinydantzler would likely know better than anyone).
[There is, however, the possibility of such a player being the only player of merit on said team, and there are a number of factors that could prevent said team from rising through the ranks due to misfortune to their NT player (i.e., injury or foul trouble causing the team to lose games). As a side note, if this team was in my league, I would play my best defenders against him, even if they were out of position, knowing the rest of his team is utter crap.]
I think England is a lousy example, though, as there are less than 250 active teams.
Assume there is a team in division III in Spain, Italy, Germany, France, Turkey, USA, Brazil, Portugal, Israel, Greece, or Argentina with a NT player. The simple fact that there are teams in division IV (and V, in some cases) in these countries means more competition on lower levels. Only 16 teams (out of 256, mind you) can leave division III for the greener pastures of division II in season. The way the playoffs are structured, the team in question could face a crunch time in the first round and be knocked out, and could possibly face repeated seasons in III.
Without being much more verbose, it is also rather likely that you are correct (if I even managed to understand you correctly in the first place).