Specifically old raids and dungeons. I know much of the content before BFA is trivial at high levels, but are there dungeons or raids that still pose a threat? Can any class solo SL normals yet? What’s the cutoff? Does Chromie time affect soloing legacy content?
With regards to efficiency and survivability, is there an overall best class/spec to solo legacy content with?
BM hunters are really good.
With trailblazer you get a 30% speed increase
Anything pre legion you can kill everything on the move from ranged so you basically just run through the instance and loot things.
Blood DK has amazing survivability but lacks mobility.
DH has great sustainability and speed.
Those have always been the three best for me.
Which spec?
Most classes can kill anything, pick the ones that can move faster instead: Shaman or Druid.
If you set up an Arcane or Frost Mage as a Legacy runner, you can really get stuff done quick.
They port everywhere, move pretty fast if talented the right way, and AoE is superb.
If you are looking for Achievement completion, BM hunter is really the way to go. The pet in many cases can act as a “second” person for those weird achievements and get them done easy mode.
DH’s (it HAS to be Havoc due to abilities and AoE and double dash) are also suuuper fast. However, anything pre-Legion I feel is limited in the sense that they can’t learn any transmog sets if they are tagged by class, even leather ones. So that kind of discourages me from using mine unless it’s mount runs.
Rogues are also very, very, very quick to run stuff. Especially in Outlaw.
EDIT:
I will add a sleeper.
Fury Warriors. They have amazing mobility with double time and Leap-w/dash and you literally just spam WW to cleave the entire place. Super fun to run with.
Edit 2: Whichever alts you designate for Legacy, be sure to grab some essentials. Like the 3 guild cloaks that teleport to Org for example (CD is NOT shared). All Wormholes if the toon is Engineer. All 3 Hearthstones. Rings to dalaran and the Tournament tabard. It can all make you super mobile if you’re not a mage.
I’d like to suggest a race too: Dark Irons. Mole Machine can drop you close to some raid entrances.
Legion M raids 
I’d be willing to wager a tank or tanklike (pet class) spec could but it’d take forever and I don’t know if it is worth it outside of the challenge of doing it.
SubLegion () It doesn’t matter, I personally prefer hunter for speed (trailblazer talent) or a stealthy for things I need to skip.
I wonder does the speed traits from BFA pieces still work? Or are they inactive already?
BM Hunter, fast speed ooc, stealth and pet to act as tank. Some fights like say Heroic/Mythic Coven of Shivarra are a pain to do without being a hunter.
With the amount of burst MM has, you could probably do MM too without much issue.
They work 
. I know it’s weird but I have a friend playing though just BfA an not stepping into Shadowlands. Their endgame is doing visions/seeing if there are any parties for heroic/normal Nya’lotha.
YES, the benthic speed set still works. It’s a stat. However the amount of stats lost vs current gear mean things “might” not get one shot.
Looks like 280% speed is still possible:
Beast Master hunter if you want a pet to tank and just kill stuff as soon they enter your range.
But i also like Elemental shaman, spirit wolve to run faster inside instance at will, chain lightning to delete pack of mobs, and a very tanky earth elemental when needed.
Warriors is good with fury.
Tanks can solo Mythic legion raids. My bear can at least.
Havoc DH and Druids of any kind due to the speed boost and good AOE 
I will second BM hunter but for a point I haven’t seen listed.
Leech
Some M raids in legion still do damage due to stacks of debuffs reaching extremely high counts. Without leech your not going to be able to dps the bosses quick enough without dying to those, even with pots and self heals
While all are capable Druid is best IMO. Cat stealth, speed bonus, bear can moonfire pull and charge pack to pack. Swipe swipe swipe.
Just want to point out you can’t access both of these at the same time, they’re on the same talent row; Trailblazer and Camouflage are level 30 talent choices.
Get a feral druid.
Take balance affinity
Get Oath of the elder druid legendary.
You now have a 17 yard range thrash.
That’s a 34 yard diameter spammable sphere of death for 1 shotting as you run through old stuff
You also have a 56.25% increased speed passively in cat form with the buffed feline swiftness.
Now get the other move speed cheese going, the +20% move speed from something like somnambulist (dreamweaver) and others.
You also get sprint and Stampede, Stealth for skipping unwanted parts… it’s pretty much the best ever for this. IMO of course.
If its old content, use a priest and spam holy nova while keeping levitation active and feathers in front of you in style.
Leviosa…