Mayan ruins - best way to see

This is a large area (spanning parts of multiple countries), and getting around takes time. I've been to almost all of the major (and lots of the minor) ruins around the region, and I think you are going to face some challenges doing what you are proposing in the way you are hoping to do it.

Because the ruins are scattered across a vast area, and because the road infrastructure is not set up for high-speed travel between the sites (some of them are actually quite remote), unless your trip is long (expedition-length), you're only going to be able to see a few of them easily, a handful at best if you are more ambitious.

Some sites are extremely popular (ie crowded). They are still magnificent, but you will be sharing that magnificence with hundreds (thousands?) of other folks coming off many cruise ships and countless organized tour buses. The vast majority of tourists only go to 2 or 3 sites, because they are famous and because they are "easy" (a packaged tour takes you there). This is not the best way to see these sites, but it's what 95%+ of people do.

Having been to most of these sites myself (as recently as next April and as far back as the 1980s) I would have to say that "doing" ruins from a cruise ship guarantees you the absolute worst possible experience there: you will be at the ruins mid-day, in the full heat and humidity, and during the peak crowding - two factors that will be hard to ignore. This is true at the biggest and/or most popular sites that I'm sure you are looking at (Chichen Itza and Tulum). Chichen Itza is the biggest and most popular site; Tulum is not big but its coastal location (relatively close to all the cruise ships and resort areas) and it's unique photogenic qualities (it's the only ruin located next to the ocean, so it is the most-photographed - and most heavily promoted - place in Mexico). I think both Chichen Itza and Tulum are worth seeing, but I would avoid being at either during the middle of the day (you know the old saying about "mad dogs and Englishmen"...). These places are best experienced first thing in the morning, literally as soon as the gates open. Charge inside, see the places for an hour or three, then get outta Dodge as the swarms of tour buses roll up. To do that, you need a car, either your own if you can manage driving in Mexico, or someone else's who will be driving you and your party around.

Location, location, location. With the exception of Tulum (perched right on the beautiful blue Caribbean, oh so photo-worthy, and oh so crowded, too), pretty much all the other ruins of Mexico, Guatemala, Honduras, Belize are inland, NOT quickly accessed from the popular tourist coast.

If you are interested in ruins, yes, you need to do Chichen Itza (it's wonderful) and Tulum (it's pretty, but as ruins go, kind of meh except for that pretty blue water in the background, there's even a small beach there you can swim at). Nearby Tulum, Coba is worth a visit, too. It's not as crowded and feels more like ruin-hopping used to be before the complete explosion of tourism in the region. Beyond those, they start getting less overwhelmed with crowds and provide a much better experience, but are not quick/easy to get to. Some are pretty remote (especially to those who arrive on the popular tourist coast and have little time/limited ability to charge off into the jungle).

The good news: there are LOTS of ruins scattered widely across this region, some as good as - or better than - the ones you've heard of that get intensely marketed and intensely crowded. The bad news: the famous places are quite crowded, and that comes with some downsides.

If you can manage it (not easy for everyone), go beyond the most famous places mentioned above. Uxmal is as worthwhile as Chichen Itza. Calakmul is better than any of them. There are a dozen places you've never heard of that are nearly as good, but have no crowds at all. Hope that helps. Buena suerte.

..."doing" ruins from a cruise ship guarantees you the absolute worst possible experience there: you will be at the ruins mid-day, in the full heat and humidity, and during the peak crowding...

We have been to many of the major and minor Mayan sites on cruise ship excursions, and have had totally different experiences from the above poster. Yes, the heat and humidity will be a factor to consider. But that will be the always be the case unless you stay close to the site and go in the early morning. The same is true with regard to the crowds.

The only downside we ever had was one excursion that involved a long bus ride there and back, and only had 45 minutes on site. But since it was advertised as having 2 hours on site, the cruise line reluctanly, after a bit of "negotiation", refunded the cost of the tour.

Since you "...prefer not to have to get ourselves from place to place too often...", it might well be worth to consider a cruise. Nothing like leaving all the logistics to someone else. And rarely do they ever stop overnight in that part of the world, so doing day trips might not be possible.

I can’t speak to Mexico but I spent many years in Belize. If you want a beach experience, you could stay on Ambergris Caye and take one of the many day trips to Altun Ha or Lamanai. The tour company will pick you up at your dock and bring you to the ruins via boat and bus. In the past, Lamanai did not get as many visitors as Altun Ha due to its remoteness. You go down a jungle river to access it. It’s been quite a few years since we went but we loved it. Check with the tour company if you want to know which days the cruise ships will be in port.

Edited to add:
You could also get a transport from the Belize port/cruise terminal to San Ignacio which is a town in western Belize right over the border from Guatemala. There are numerous Mayan ruins tours from the townn including Tikal.

You won’t be able to see everything no matter how much time you have, but here are two suggestions for a good base:
For Mayan, Merida in Yucatán. You can day trip to the two major sites of Chichen Itza and Uxmal, plus there are many smaller sites in the area. And Merida is a very nice Mexican city. Or you could combine Merida with the pleasant smaller city of Valladolid, which is a bit closer to Chichen Itza.
You appear to be open to “other”, ie other than Mayan, sites. An excellent trip would combine Mexico City for the Aztec Templo Mayor site, day trip to pyramids of Teotihuacan, see the anthropology museum (best collection of Mexican antiquities in the world), then a short flight (or longer bus ride) to beautiful Oaxaca for Monte Alban (as impressive as pretty much any of the Maya sites) and Mitla.
Of the places I’ve been in this region, I think the most impressive is Tikal in Guatemala, but it’s far from anywhere else.