We don't want suck characters with all stats less than 7. We don't want to be forced to play a fighter because we didn't roll even a 10 for INT.
Keep that in mind. It's worth coming back to.
But we also don't want cookie-cutter PCs that were just using the exact same point buy as the last PC.
Why? Why in the world would this be an outcome? What's forcing you to arrange your points in the same manner each time? Of the 15 different PC's the players have created so far in this latest campaign, I doubt any two shared the same stats. I'm fairly sure none have shared the same class or combination of classes. If you don't want cookie cutter builds, why make them?
Celebrim's misinterpretation was that I wanted wildly random results with low numbers. I want random results in a certain desirable range.
Except 4d6 doesn't give you that. It gives you wildly random results. You can still get 3, 4, 5, 6, or 7. It doesn't guarantee results fall in a certain range. It makes sucking less likely sure, but it doesn't prevent it. It just makes you think, "Well, it won't really happen to me." The problem with 4d6 take best three is sooner or later you get it producing results that look like 3d6 and then your desired results fail. I've seen it happen. I've DM'd enough players to know that look on their face when they roll up the character and they got the equivalent of 18 point buy (or worse) - which is going to happen if you've got 8 players at the table and you make them all stick to their rolls. Sure, I've played with 7's. I played Ogden Mudstump, the dwarf thief with 5 charisma.
I just rolled up 4d6 take 3 in order, it was: 10, 5, 10, 11, 6, 8. Second was 15, 12, 17, 6, 10, 7. Third was 7, 8, 13, 5, 15, 11. Fifth was 12, 13, 15, 4, 13, 13. Sixth was 15, 9, 12, 10, 13, 13. That's what 4d6 really looks like. Random. Actually freaking random. You take some random samples, and you'll get anything from like 9 to 40 point buy.
What if I told you that we were going to use point buy, but we were going to use 5d10 to determine how many points you got to spend?
Tthat guy with the 10, 5, 10, 11, 6, 8 is not going to play it. The dice didn't 'do their job' - though in 30 or 40 rolls, probably 4 or 5 will be that bad. Sooner or latter they are going to end up playing the seventh set of dice I rolled while writing this: 15, 15, 17, 13, 15, 13. That's the thrilling sort of results they were gambling on all along. If you have them roll away from the DM, you can bet on them forgetting those first few sets. If they roll in front of the DM, you can bet they'll beg or kill the character off. Random means: "Either I win the jackpot, or I'm going to enter into a complex negotiation that makes a mockery of the so called rules of character creation."
If what you really want was no numbers less than 7, why don't you do 2d6+6 straight up in order? If what you really wanted was results in a certain desirable range, why don't you use point by with 22+2d4 points?
I would argue that the things you prioritize aren't in the these alternatives, and they aren't 'don't get low numbers' or 'don't have no stat higher than a 10'. I played that way about 15 years, both as a DM and a PC. I know some of the attractions. Rolling your stats seems more real, more hard core, especially if you are used to rolling your stats. Point buy brings that gambler's thrill - maybe this time I'll get lucky. Point buy brings truly odd results - two 17's and 3 9's, an 18 and 3 8's, 2 18's and a 4, etc. - that you might never otherwise consider playing. And then there is always the interest of having a pretty decent roll in a stat you'd never otherwise place it in, which arguably is my favorite part. That randomness has some benefits, but... it just isn't worth it not the least of which is because in practice its more like character generation in Baldur's Gate (the video game). People roll until they have the numbers that they can live with.
If what you really want is inspiration to play something you wouldn't consider otherwise, go ahead and roll that 4d6 keep 3 six times - then use point buy to match the result as closely as possible. I'm done with using fortune rolls to determine the outcome of character generation. Save the fortune rolls for the game.