My preference- and what I use in my campaign- is "Everyone Starts at First Level."
I think the general practice of starting new PCs at level 1, which kind of worked in 1e and 2e, got savaged by the tighter (and I'd say overdesigned) balance of 3e and 4e. It would probably work much better in 5e than it did in either of its two preceding editions.
I'll testify to that. "Everyone Starts at First Level" (ES@1) was my rule in my campaign from pre-1e to early 3.0, when I realized that it simply wouldn't work anymore. In 4e, it was even worse- the "+1 to everything per level" bonus saw to that quite thoroughly!- but 5e, by flattening the math and using bounded accuracy, has really made it possible again.
So, for the sake of conversation. Why level 1? Why not level 2? Or half the party average? Or one below the party average? Or just the party average?
There was a whole gigantic thread on this recently, of course. My answer is that using ES@1 means that each pc has a significant history, has earned every single thing on their character sheet, the player knows exactly what kind of stuff the pc has encountered, and so on. Nobody who plays the game feels like the new player gets freebies they didn't. And so on and so forth.
Obviously, ES@1 isn't for everyone, but it works for me and my groups.
Moreover, once the party is high level, why do they take this relative weakling into situations in which they need to depend on each other for life and limb? If you were playing basketball, and one of your star players sprained an ankle landing after a jump shot, and had to sit out, would you then take the weakest player on the bench to replace him? Why aren't the PCs seeking out someone more on par with their own abilities, who won't need babysitters or extra protection in order to survive?
If you look at both history and fiction, most of the time, great heroes are accompanied by lesser figures. Apprentices learn at the side of their masters. D&D has a long history of henchmen, cohorts and followers coming in alongside their masters; why not associates, friends and allies, not all of whom are as bad-ass as the top level guys?
Think of Arthur and his lesser knights; of the group of heroes in Sailor of the Seas of Fate, which had a handful of epic guys and a bunch of warrior-follower types (IIRC); the Fellowship of the Ring, which included both the low-level hobbits and the high-level Legolas and Gimli and friends; and so on. Don't get me wrong- I totally recognize the differences between a story and a game (e.g. plot immunity), but you'll see the same thing in the old sagas and tales of myth and history (Napolean, Marius and Caesar led many soldiers lower-level than them; Beowulf had his group of men with him when they want to Hrothgar's Hall). And one of the things that is at the heart of D&D, IMHO, is the notion that you can emulate those old sagas and histories and those bits of fiction.
Obviously, YMMV, and ES@1 is clearly not for every group (and perhaps not even for many). But for me, and for some others, it isn't just a good way to go, it's downright wonderful.
(I'll add, tangentially, that some early D&D knock-offs, such as Arduin Grimoire, actually presented alternative hit point models specifically to encourage this sort of thing.)
That's not even to address the issue of how do you know a dude is a high-level fighter vs. a low or mid-level fighter on sight- I get the notion of running tests, but sometimes adventurers meet in the wild and have the "Hey, pc, join the party" conversation smoothly and easily without any of that kind of thing. So, you know, there's that, too.
FWIW, in our session last Saturday, we had 3 4th level, 1 3rd level, and 1st level PC. It wall seemed to work fairly well.
Last night's game we had 2 pcs at 4th level, 2 at 1st level (and only 9 xp from level 2!) and one pc at 3rd level at the start. By the end, we still had 2 pcs at 4th level, we had 2 that had juuuuust hit 3rd and one at 1st (who replaced the 3rd level guy after he was thrown in chains by the army for murdering some children). It was a long session- around 8 hours- with a lot happening in it; everyone was challenged, everyone was useful and nobody felt like they couldn't contribute.
I will absolutely grant that this may change as the high level pcs grow in level; but in this campaign, there's a good chance that everyone will eventually be playing multiple pcs (though usually one at a time) on different adventures, so groups can kind of self-select their composition and challenges. So that should help manage the issue, if it proves to be one. But don't forget that a party with high-level members can still be challenged by goblins and orcs in 5e! Just because the party has an 8th level dude in it doesn't mean that the pcs will be constantly facing CR 8 monsters.