If the normal (non-tubeless) tyre holds air and sits firmly on the bead, are there any problems with running without a tube? This particular setup is with a Continental GP5000 (tube tyre model) and a Hunt tubeless-ready rim. The tyre went on very tightly and seated into the outer hooks with some loud audible pops. Seems solid, though I am sure the manufacturer will say it's not recommended. Has anyone tried something similar? In particular, why would a tube make the system better/safer (after first inflation)?
This used to be commonly done for MTB or cyclocross/gravel, where volumes are bigger and pressures lower. I, OTOH, run a tubeless gravel tyre on a non-tubeless rim quite happily.
For road bikes, with low volumes and higher pressures, it is generally considered more risky. That does not mean that your specific rim and tyre combination will not run OK in practice, but the risks are much higher than with the high-volume types. A decrease of pressure in a corner on a road bike with the low-volume tyres can be quite dangerous, while for MTB an occasional burp of air is not something unheard-of and often will result only in a delay in a pit stop.
You cannot safely run a non-tubeless tire as tubeless. If the tire does not blow off the rim immediately, it could blow off while you are riding. If you compared weights between tubeless and non-tubeless versions of the same tire, you'd see that the non-tubeless tires are lighter. I am not sure how much of that weight differential goes into the casing versus the bead, but I think it's at least some in both. The beads of tubeless tires are designed to form an airtight seal with the rim. Non-tubeless tires don't have to do this.
Vladimir is correct that in the past, you would see people running MTB and cyclocross non-tubeless tires as tubeless. Air pressures in these tires are a lot lower. For example, I typically run my 25mm non-tubeless road tires at 78-79 PSI. I would run a cyclocross tire under 30 PSI. At road pressures, the risk of an informal tubeless conversion is not worth it.
Rim tape, lots of it is your answer. After that, when you have achieved a tighter seal between the non tubeless tire bead and the rim because of the tape then lots of sealant is the next step. After that though there won't be much difference in weight savings if you had just used a tube. But ride comfort is still a bit better. But if you have a full suspension bike then you should have just run the tubes and tweaked the suspension for the same feel.