A few thoughts:
- Bring a large base plate for your car jack. If you get a flat on the road, chances are you will pull over to the side and onto dirt. A thick, wide base plate will help spread the weight of the jack and stop it sinking into the ground. I carry a couple of pieces of 50mm (2 inch) think off cuts; others may use a steel plate with bolt holes to lock the jack to it.

- Even if you don’t know how to use all the tools and spares you carry, it is still worth caring them because the person who stops to help you may know. Manufacturers are all different, I may not have the correct size tool for your car, and items like engine hoses; fan belts etc are very specialized sizes.

-There is also a sort of etiquette to offer a Good Samaritan use of your tools and equipment so that they don’t have to unearth theirs or get them dirty / damaged; having said that people often prefer to use their own tools, but its nice to not assume that they will have the tools / parts or that they will necessarily want to use their own. (On a trip, some of my tools can be buried quite deep in the back and its a job to extract them all. If its raining then this can be extra inconvenient).

-Compressors: There are lots of cheap compressors on the market, but if you have big tyres (eg 4WD, or even some large family sedans) the cheapies may not actually be able to pump them all the way up, and they will be slooooow ! An additional consideration is whether you intent to ever reseat a tyre on a rim. To do this you need a compressor that can deliver a large, high pressure volume in a short amount of time. Basically if you just want to pump up a small tyre, then a quality cheapy is probably OK, but if you have bigger tyres and want to reseat tyres, than you need a bigger, better, more expensive unit ! Activities like 4WDing sometimes necessitate lowering and then raising tyre pressure; with a slow compressor this can literally take hours – not so good if you are with other vehicles and they have to wait !

-Chains: I think its been covered, but chains are generally used for dragging or extending the reach of other straps or cables. For instance we use a chain to drag fallen trees off tracks. It is very abrasion resistant, so even if we have a winch, we put the chain around the tree so we don’t damage the winch cable.
A snatch-um strap is used to recover a vehicle which is stuck. This is a webbing strap with a 20%-30% stretchability and a recovering vehicle can pull it and use the kinetic force to gently pull a bogged vehicle; they are very effective. These can be used as a tow rope and work well, but are quite expensive, so often people use a similar, but less elastic strap. Because the straps are rated for only so many recoveries, and using then can get them very muddy, it is once again etiquette to offer your own and not rely on others to use theirs or muddy it – unless they offer.

Finally consider what you would need if you become stuck in your vehicle for an amount of time ? What if you had to leave your vehicle ? – leave a message on the dash; food; water; shelter; warmth; signalling. Think Car survival kit.

Some thoughts regarding tools:

Tires are usually put on with impact wrenches nowadays and are usually pretty hard to remove manually. I like to carry the appropriate impact rated socket, a breaker bar, and a pipe extension.

Car Toolkit: I try to carry inch and metric sockets, ratchet, spinner handle, (some torx, hex, phillips, flatblade, and socket insert bits that can be used with the spinner or ratchet) some cheap open ended wrenches, 5-in-1 Screwdriver, Utility Knife, Wire Cutters, inch/metric allen keys, needle nose pliers, vice grips, magnetic pickup, ballpeen hammer, mini-hacksaw, wonderbar, bright led flashlight, gloves, cloth rags, electrical tape (scotch super 88), gorilla tape, a container to hold parts in (usually use the cup the electrical tape comes in). This is all jammed into a small gatemouth toolbag. I have a safety orange tarp with the plastic sleeve it was shipped in (either can plug a shattered window in a pinch or pad your work area on a wet roadside), along with a wool blanket---these line my trunk. I carry an e-tool, estwing camp axe, and a gerber folding saw as my "expeditionary tools", though I supplement the e-tool with a fullsize snow shovel during the winter.

First time I ever bought new tires I went back for the free rotations. Then needed to take one off for something and it too two of us standing on a pipe on the lug wrench handle to get the nuts loose. I bought a torque wrench and from then on I just rotate tires myself.

I was in a parking lot jumping up and down on my lug wrench to change a flat not all that long ago. It occurred to me after the second time I fell that my wife probably would not be physically able to change a tire in a similar circumstance.

Later, I cut a couple lengths of galvanized pipe and stashed them with the spare in her car and mine. Hopefully more than doubling the lever arm of the lug wrench will allow her to break the lugnuts loose without hurting herself.

that's the worst part. i am a tire-changing pro, as long as i get the lug nut loose.

Yep, have those myself, but even with them a lug nut tightened too far you may not be able to get it off. Tightening your lug nuts to the proper torque goes a long way towards helping with that.

i ran this photo before--


the old Plymouth i bought for just hauling the canoe around had lug nuts that were just about impossible to get off with anything but a cross wrench and a pipe extender.a flat in one of the parking lots up at the wilderness canoe park or along the back roads on the drive up would have been a major hassle.i removed all the nuts and replaced with a drop of oil and a torque wrench.i also replaced the spare with a real tire so i could drive all the way home on it and not have to limp into a "end of the road" town for repairs and find the one gas station closed.
and by the way i never put that fire hazard propane and heat drum into it over the winter.i never drove it anyplace i could not walk home in a half hour.

coat the threads with "never seize"

You have to be careful, the torque is spec'ed with a dry bolt in mind. If you put anything on it, never seize, oil, etc you will need to dig out an engineering book and look up torque tables and adjust accordingly. You'll probably have to look up the existing torque in the chart then look at all the other inputs then look up the corresponding based on what you put on the threads.
If your not careful you could end up loosing the lug nuts, if a lug centric wheel break the lugs off with.

Speaking of lugnuts. It's not a bad idea to actually try changing a tire on a new (to you) vehicle with only the tools you'll have with you on the road.

I discovered with our 12 passenger van that each wheel has eight lug nuts and the nice long lug wrench that comes with it is too long to spin a full 360 degrees without hitting the ground. If you can't spin the nuts with your fingers because the studs are a little rusty, it would take hours to change a tire.