That far from destroys competitive strategy. If anything it promotes it. That way picking instead of just changing characters to a counter pick you have to develope new gameplay startegies.
Ex: You play ICs. You are going through the tournament and you can't pick a character based only on the fact that it counters any one character then so you choose who you are best with. You come across a Peach. Oh no! What does any and every IC then do? Switch to Marth. Wow, real creative and strategic. Now you don't even have to have a strategy just a grab/f-smash happy Marth.
Now this is something that is planned for. The chance meeting of the Peach was just dumb luck for you because they or you could've lost before or the seeding could have been different etc. Thus you now have to figure out how to battle Peach with ICs.
Now you're thinking, "Well based on my programmed ways that are always right and must be so because MLG said so then the Peach player now has to learn to play against Marth."
Wrong. The Peach will likely suspect this and change themselves unless they feel Peach is still their best bet. They will merely wait and counter-pick you next match. Any motivation to get better in a certain match-up is either gone or dulled.
You have to play the match-up, and you know it will likely come up, so you have to develope strategies and techniques specifically for the fight as opposed to waiting to counterpick yourself.
Now your thoughts on saying that people that are incredibly good with many characters are better (and thus deserve the advantage of using them) than those who are good with 1 or 2 characters is correct though your reasoning makes no sense. By the way, this very thought is why I have always thought that Azen is better than Ken even though Ken wins more often. But it also goes that you don't want to be a jack-of-all trades, master of none as you will get nowhere. So like you said, they have to be quite good with the characters.
You say this gives the random low tiers a chance. This is completely illogical. If the random low tier has a chance then the other player will merely switch to one of the many characters that counter low tiers (the reason they are low tier for if they didn't have many bad match-ups they'd be really good to use). The old BIN style gives low tiers a chance as there is no counter picking and if you happen to come across a bad match-up you have to or already have, assuming this has happened before and you've worked on the match, developed a playstyle to help even the playing field.
This would open up the tournament scene to more players and thus more spectators and interest. We'll use good ol' BUM as an example.
BUM's DK is freakishly good. Now some of those Fox mains, being forced to stay Fox, may decide to switch mains to ICs, Sheik, perhaps Falco, or some other character as DKs now have a good chance against Foxes instead of relying on the counter pick crutch (for that really is all that it is) for a match-up. This promotes diversity of characters amoung tournament goers and now people who don't go to tournaments will because the scene is more open (diversity = fun and inviting because it is not monotonous and it give low tier players a better chance... plus exclusitivity breeds insecurities in others). I know this is the case because I have several friends who don't go to tournaments because of this very reason and it is a shame because they are very good but happen to use low tiers and it is a big reason why I don't travel far to go to tournaments.
Some may be thinking, "But then everyone will just main Space animals because they have the least number of bad match-ups."
Nope because if everyone is using the same character than one person with a main that counters them will easily win the tournament. "But then you just need to develope strategies to beat that character!" I know, read up a few paragraphs. But when others come along and that go well against both then so on and so on people will switch it up and we start arguing in circles!
All that counter picking is is a crutch for players to lean on so that they don't have to be as good with a character because they know they can counter pick.
The current style promotes diversity in the top tiers almost exclusively but the other spreads that around to all tiers a little more evenly.
That being said, the current style isn't bad. There's no reason the other is bad as well though. In fact I think tournaments should have different rule sets as this best promotes fun and competitivness as you are forced to constantly adapt and it changes things up a bit to keep it interesting. Obviously a series like MLG or a regional circuit can have the same rules to see who is best at a given style, but again there could be another series that changes it up every time.
MAIN POINT!!!!!! ----------> I think we, as players, should push the sport and try to open it up to new people and expand it and I think this type of diversity amoung tournaments would really help to accomplish this for the reasons mentioned above.