It's not a bad idea to bring a spare tire instead of a whole wheel, but I wouldn't think that would be for "trailside repairs" (at least not reliably). It can be really hard to change a tire on a motorbike wheel with irons on the side of the road, even for someone who knows what they're doing. I imagine a car tire would be a lot harder, potentially. For bikes, a lot of it depends on the tire compound -- I've had a TKC-70 that I couldn't get on if my life depended on it with irons, but I regularly swap out my K-60s by hand. Similar tires in terms of function, on the same rims with the same tools and technique by the person doing the work, but for whatever reason the K-60s are much easier to change.
We never bring two tires except for our trip to Alaska, and I think our reasoning reflects that there is a third part to the equation of "two tires or one" that I've not seen mentioned yet -- lots of folks have talked about whether or not you need a second spare in North America, and the consensus so far in the thread seems to be "probably not". But I would say that the third factor is time -- what kind of a trip are you having? Do you have time to wait around for replacements that might be a few days away?
For example, on our Alaska trip, we had to hit an average of 600 kms per day, every day, in order to make the trip happen with our timeline. That's hard mileage when camping, especially on a bike (My wife and brother were in the 4x4, we convoyed up together because I really like riding). If we had to park up in Deadhorse and wait a few days for a spare tire to be shipped in, it would have thrown off our plan in a big way. However, with a second spare, we had options -- let's say we shredded a tire on the way to Prudhoe. With one spare, we'd be forced by pragmatic thinking to stick it out in Deadhorse until the spare was replaced/repaired. However, with a second spare we would be able to return south to Fairbanks with some peace of mind that if we shredded another tire, we still had another spare. And we could get any tires we needed fixed in Fairbanks within a couple of hours (as opposed to waiting a few days in Deadhorse) and thus that would be less impactful on our overall schedule/trip. We also brought a tire repair kit but sometimes, tires can't be repaired so we liked this extra redundancy.
As it happened, we didn't get a single flat in either vehicle on that trip. In fact, I've only ever had one flat tire, and it was in a parking lot in a major city. But, if I were to do the Alaska trip again, with the same time crunch (or a similar remote trip with a time crunch), I would probably bring a second spare just in case assuming I had the payload and storage to do so.