Are tubeless tyres an upgrade from tubed tyres, or just different? Bicycle tires have reached a good degree of practicality with clincher tires as used with inner tubes.
Unless riding over sharp objects that have a tendency to puncture the tire, punctures are extremely rare. When an inner tube is punctured, most of the time the damage is limited to the inner tube, which can be repaired to be nearly as good as new in 99% of the cases with glue-type patches. The cost of such a patch is a fraction of a dollar. In very rare cases, the outer tire is damaged necessitating sticking an adhesive tire boot to it as a temporary solution and replacement as a permanent solution. Most outer tires do not die of damage but rather of old age, i.e. worn tread. Thus, the fact that a tire boot is only a temporary repair and not a permanent one is not a big monetary issue.
Inner tubes with factory-installed sealant &quot;slime&quot; are available, but because of the rarity of punctures, such &quot;slime&quot;-type inner tubes have not reached major acceptance. Nevertheless, the slime tubes can be found for sale for those who absolutely hate patching tubes.
However, nowadays tubeless tires seem to be arriving. A true merit of tubeless tires is avoiding one type of user error: pinch flats. The user error requires the presence of three conditions: (1) the tire must have a narrow cross section, (2) the tire must be underinflated, (3) the rider must go over a bump at too high speed. I have recently not had pinch flats because I try to avoid every one of these conditions.
However, it could be argued that this merit of tubeless tires is false sense of safety. When one learns to ride at high speed over obstacles with underinflated narrow cross section tires, the probability of rim damage increases. Thus, the possibility of pinch flats (that can be repaired easily with lightweight tools you carry with you) is very good in preventing the possibility of rim damage (that might be tricky to repair at roadside).
As I understand it, tubeless tires are often (always?) used with a sealant &quot;slime&quot; that seals some of the punctures. However, I have some suspicions about the merit of tubeless tires:

What if the puncture is not adequately sealed by the sealant? Can I patch a tubeless tyre in a permanent way with a glue-type patch? A permanent repair without a slow leak for inner tubes is easiest when you always carry one spare tube so that you let the glue dry before using the newly patched tube. Unless some major form of technology breakthrough has occurred of which I'm not aware, if a tubeless tyre can be permanently repaired at all (which I suspect won't be the case because of the sealant &quot;slime&quot;), you should always carry one spare tubeless tire with you to let the patch glue dry. Carrying a spare tire takes more space and weight than carrying a spare tube.
What if the cords of the tire are damaged? With a tube-type tire, there is no sealant &quot;slime&quot; so I am certain that an adhesive tire boot will stick to the inside of the tire. However, with tubeless tires, the sealant might make it more difficult to ensure the adhesive tire boot stays in the intended place.

So, why should I care about tubeless tires? I see them as a non-repairable solution that harms the use of emergency tire boots and at the same time encourages the rider to cause rim damage.
What if the puncture is not adequately sealed by the sealant? Can I patch a tubeless tyre in a permanent way with a glue-type patch? A permanent repair without a slow leak for inner tubes is easiest when you always carry one spare tube so that you let the glue dry before using the newly patched tube. Unless some major form of technology breakthrough has occurred of which I'm not aware, if a tubeless tyre can be permanently repaired at all (which I suspect won't be the case because of the sealant &quot;slime&quot;), you should always carry one spare tubeless tire with you to let the patch glue dry. Carrying a spare tire takes more space and weight than carrying a spare tube.

No, if the sealant and specific plugs (mentioned in Diado's answer, the same ones that are used for automobiles and motorbikes) fail to seal the hole, but the hole is not extreme, you can just swallow your pride, insert your spare tube, and carry on riding. If the hole is large, you will need a tyre boot in the exactly same way you would need it for a normal inner tube tyre. Then at home, after a good clean, you can indeed use a tube patch or one of the larger tyre-specific one. If the hole required a boot, it may also require some sewing. and then covering by a polyurethane glue and a patch.

What if the cords of the tire are damaged? With a tube-type tire, there is no sealant &quot;slime&quot; so I am certain that an adhesive tire boot will stick to the inside of the tire. However, with tubeless tires, the sealant might make it more difficult to ensure the adhesive tire boot stays in the intended place.

That is unlikely. Quite the opposite, it will make it stick. Or at least be neutral.

So, why should I care about tubeless tires? I see them as a non-repairable solution that harms the use of emergency tire boots and at the same time encourages the rider to cause rim damage.

It is repairable (I repaired several punctures myself) way that makes better riding properties especially off-road (lower pressures) and reduces the number of punctures you have to deal with (no pinched tubes, small thorns are plugged). Small unsealable holes can be plugged without taking the tyre or indeed the wheel off the bike.
Any potential rim damage is the responsibility of each rider, their selection of the tyre pressure and their choice of their riding style.

The user error requires the presence of three conditions: (1) the tire must have a narrow cross section

Pinch flats can also happen with MTBs.

(2) the tire must be underinflated

Low pressures are selected for good riding properties and it is impossible to perform well in certain off-road events (cyclocross, cross-country) with such high pressures an inner tube would require. The other option is a tubular, but that is even more hassle.

(3) the rider must go over a bump at too high speed.

Stones and roots may be hidden by mud, leaves or appear suddenly behind a corner on a trail. Sometimes they cannot be avoided even on a maintained course, less so in unmaintained trails.