I think it will depend on what you are going to be pulling on what to bring. Of the times that I have pulled parts I usually take the basics (Metrics or SAE wrenches, sockets, screwdrivers etc.) including a hammer, pliers and wire cutters.
I once found a leather steering wheel I wanted and did not have a steering wheel puller but was able to hit the steering shaft as I was pulling up the steering wheel to get it off
I use a 5 gallon bucket to carry my tools. Easy to throw in extra bolts/fasteners etc plus if you bring long extensions it is easy to place in the bucket and not have them fall out. Sometimes the yard staff wants to look at your tool bag first so a bucket makes it easy for them to see what I am bringing in and pass me through. Last time at the auto yard I pulled a set of Vortec V6 heads for my boat motor. For a pry bar, I locate an old lug wrench in a car an use it if I have to pry up something (Vortec heads off of the block) then I usually just bring it out with me. Sometimes I have been charged a dollar or two
The bucket can be flipped over and used as a seat if pulling brake calipers, suspension pieces etc. You can also stack 2 buckets together. One to carry your tools and the other to carry the part(s) to check out.
Hope this helps
at the jyards I goto there's always wagons/wheelbarrows to haul your stuff around. I've seen buckets used for tools & parts but not used as seats.

And I've also got a 12" pry bar sticking out of the largest canvas bag.
What you bring totally depends on what game you hunt.
I have found that Tekton sells a spiffy 1/4" metric socket kit with ratchet and spinner handle. Bring a set of bits and bit ratchet. Side cutters, and good utility knife. I've pulled a lot of unobtanium for 30-40 year-old cars with just this.
I do like panel load backpacks and tool bags with shoulder straps.
-Ryan
since I don't live that close I spend a full day at the jyard and just bring a full set of tools. if I know I'm looking for something special, like a steering wheel or close-ratio power steering pump, I'll bring a small kit.
don't need anymore tools, just want to see if there's any reason NOT to bring only 12 point sockets and not 6pt. I don't use them at home so wouldn't necessarily mind losing them at the jyard.
found a big cheap plastic handled Ginsu knife at the jyard once, that went into my tool bag. good for cutting hoses, small wires, and other things.
panel load backpacks? not familiar with that one
I take what I know I'm going to need. Beyond that, I also always take a Philips and flat blade screw driver, a pair of vice grips, a pair of dykes and a razor blade. And one adjustable wrench.
see my response to VolvoRyan above; I bring everything since I don't know what I'll need. I also bring, as you mentioned, screwdrivers (in short and long lengths, Phillips and flathead), channellocks, vise grips, needlenose, crescent wrench, and dykes.
I pretty much follow the OP's and previous posters' plans, but too many tools gets heavy. A bag with a shoulder strap is essential.
As said, a 5 gallon bucket is great, especially for oily things.
I research what tools I need for the parts I'm getting. I bring my worst HF or no-name tools.
Some basics are universal: 10mm socket, combo screwdriver, plastic push pin pry tool, panel pry tools, rags, etc.
I bring my impact driver too. Extra "maybe" tools are left in the car. You never know what treasures you might find.
You don't need a puller to remove a steering wheel.
Too easy to lose something or have it stolen if you get distracted. A friend had some tools stolen after he left his tool bag unattended for 2 minutes. Mainly a problem in crowded salvage yards like
Victory Auto Wreckers.
Scavenge small stuff like fasteners, screws, nuts, bolts, light bulbs, clips. They usually don't bother to charge for them and they are lifesavers down the road.
10mm sockets, yeah, the little damn things just flip out of your hands and go flying. when I go to a pawn shop, I dig thru the loose tool bin for any 10mm.
I've also brought a roll of paper towels for cleaning my hands and the filthy parts.
I scavenge all the small stuff because I've had vehicles that had specific relay #'s and they want $20+ in the store while they charge a buck or two each at the jyard.
the reason I bring my drill, aside from quickly removing screws,etc, is that if you find a nice pair of power seats (like in a sports car) you can't move the seats without a battery, and you have to move them in order to reach the hold down track bolts. this is also why I carry a small 20' roll of speaker wire. use the battery in your drill to move the seats back and forth to give you access to the bolt heads
I've left tools in the parking lot, but since you can't take your cart out there and do you really want to leave it unsupervised? since I go with a friend that's not an issue but it still ***** having to walk a quarter mile back out and then a quarter mile back in.
I would add a sawzall and at least one pair of vice grips.
I've never needed a sawzall yet since I'm not cutting off body parts, but if I did then I'd also bring at least 3 4aH batteries because I know how they chew thru battery power.
for nasty filthy stuff, especially the wet stuff, I'll usually grab a trunk carpet liner and lay the stuff on that in the wheel barrow.
so the question still remains unanswered, is there any reason I shouldn't be taking only 12 point sockets to the yard and leave the 6point at home? and is there any reason to take all drive sizes (1/4, 3/8, 1/2) instead of just one or two, since I have the full run of socket sizes for each drive size?