This is kind of like what I do: there's a "floor" level at which new characters join in*, which slowly rises during the campaign as the party's average level rises. Right now my groups are in the 6-10 range and, depending on specific party, the floor would be either 6th or 7th. Stats and hit points are rolled. Starting wealth/gear is also somewhat random, to reflect the relative good or bad luck this character's had thus far in its [unplayed] career, but whatever gear is given is always suitable for the character to use (e.g. if my dice try to give a 6th-Wizard +2 plate mail it gets a +2 Ring: Protection instead) again reflecting the choices said character would likely have made.
That said, they've all got so many characters now that truly new PCs are few and far between; they just cycle the ones they have in and out.
* - exception: for a new player joining the game the
first PC comes in at the party average, rounded up. If that one dies or retires, the next comes in at the floor level.
You might be a bit surprised on this one.
We've always used rolled stats, and have rolled up and played hundreds of characters over that time. I still have nearly all the character sheets of the characters that were in my campaigns, ditto for the other main DM in our crew.
Just for kicks a few years ago I took a sample of 50 or so random characters whose careers didn't last long - "one-hit wonders", we call these - and another sample of as many characters as I could find who'd had long and fruitful careers (using 10+ adventures as the benchmark for "long"), and ran some comparisons of their racially-adjusted** starting stats.
The difference in average stat (i.e. add them up and divide by 6) turned out to be fairly trivial between the one-hits and the long-term types. What made a much bigger difference stat-wise was having at least one really good stat as opposed to lots of mundane stats (e.g. a 17-11-11-11-11-11 avg 12.00 had a measurably better chance of success than 14-14-14-12-12-12 avg 13.00 even though the second character's average is a full point higher). That said, there were one-hits with averages over 16 (!) and long-termers with averages below 11, so read into that what you will.
** - I used racially adjusted because to try and reverse-engineer all those characters to what the dice actually said would have been beyond tedious. What I was after was the stats the characters actually had in play, and we've always used the same roll-up system of 5d6 drop 2.
Conclusion: while numeretic stat disparity might make a difference in day-to-day play at the table its effect on the overall career length (or life) expectancy of any given character was surprisingly minimal.
Disclaimer: this is all using characters from our modified 1e system, where bonuses don't tend to start until 15 (but conversely, a stat of 7 never carries a penalty). In 5e, with its linear bonus progression, even a tiny discrepancy in stat average might have more of an impact; but someone else will have to run the numbers for that.
5e also has a
much quicker ASI progression than our system does.