A long long answer will follow
and some maybe not completely on topic.
I think it depends on where you live. In my country, there is no law for diving, so everybody can go diving without a cert and, you are insuranced on the national health insurance. But we don't have caves in my country. But things as 'you are diving outside a certification limit' does not make any sense in my country. Yes, the pointing fingers are there, but as instructor we can only advice people to take courses, but we cannot do anything against people who are doing things on their own. And yes, I also did before I was instructor. I have done deep night dives as open water diver, started with decompression diver after getting my aow, took some not certified divers on a dive when I was dm, and even gave an introduction of cave diving in an old slate mine in Belgium. This was all before I became instructor. And nothing was against the law, you could not get a fine. Maybe if something went wrong there was something that could give a discussion, but I know from others that the problems are bigger if you do something wrong as instructor than that you are no instructor and teach the same things. But you also can found your own agency in my country. I know from some divers who did it and bombed themselves to instructor for everything. I decided after 3 years to follow the normal path to become instructor and did it with an (w)rstc agency and then did a crossover also to cmas because I am member of a cmas club, but don't teach cmas a lot.
In France, caves are free to enter. So people get in without a cave cert (and also 1 of the rescueers of the thai cave rescue did not have a cave cert).
Now I would not advice to take someone in a cave if you are not an cave instructor. It has nothing to do with that I want to earn money, but I have rescued 2 times people from a cave (both times alive as they found an airbell). In 1 time it were no cave divers, but a group of people who did their sundaymorningdive and did not use lines. In the other case it were certified cavedivers and after analyzing, nobody did anything wrong, but only some bad luck in zero viz (and an example that bump and go is less safe than touch contact).
But yes, I have some problems with some certs in diving that are made 'mandatory' by some people. I have sometimes problems with the diving industry as with every step you must be taken by hand and there is no way for autodidactical divers. And I was one of that autodidacts with a lot of things. I always say there are usefull certs like open water, advanced nitrox, a cave diving course. There are courses just for fun like biology, most times photography specialties, etc. And there are courses that can be useful. These are certs that are in the range 'can be good, but not for everybody needed'. For recreational diving it is for example drysuit diving, but also sidemount diving is 1 of that certs. For technical diving multiple stage cave diving is such an example. If you are a full trimix diver, you are used to more cylinders. The theory behind multiple stage is no rocket science. Also the expedition trimix diver is a course that can be usefull, but not needed. A full trimix diver has no limites anymore. My instructor trainer when I did the full trimix instructor course sometimes teaches the expedition trimix. As I have never done that course and have been to 135m and 123 in a cave (these are depths outside the expedition trimix also), we talked about what I could learn from that course. He said nothing. It is the most dangerous course to teach. People who want to dive over 100m and are capable will do it themselves. Others take a course and that are not the most capable technical divers. The expedition trimix course is not written to reach more depth, but also to think about expeditions and safety divers, etc. And people who go beyond 100m most times get the knowledge by reading, using internet, asking people, or just role in it because of they belong to a group with that experience, etc.
But about the 'can be usefull courses: If you are unsure, take a course. If you have a good buddy or camera, you can do it yourself, with sidemount all skills are shown on youtube, gas is gas. But adjusting an harness can take a long time and sometimes you don't feel things yourself. This is where an instructor or good buddy or a camera can help. And yes, self teaching results also in 'diving with sidemount equipment' instead of diving sidemount, but it is not directly dangerous. A drysuit can be dangerous, but it is no rocket science. Adding more cylinders in a cave is often done carefully by progressing carefully after a full cave course. IANTD offers stage dropping in the (technical) cave course, TDI not I believe (at least we did not do it when I did my tdi full cave course). But when you are a trimix diver, you can go as deep as your cert is after getting full cave certified.
I dove already a couple of years with ccr in a cave and I cannot say I learned from an oc to ccr cave crossover. We had a lot of fun, for sure, but a ccr cave crossover does not bring you with scooters far in a cave to depths over 75m, that is beyond every course, and that where the dives we already did. But I have sometimes people for the crossover that are not that experienced and they like it to do. First the ccr cave course was more or less optional if you had already oc cave and a ccr cert. But now it looks like it changed to mandatory. If you ask people here on Scubaboard, some will find the crossover usefull, others just something that someone wants to earn money about. The answers most times depends on the experience people already had before doing it.
For wreckdiving, the full cave card is higher than a technical wreck diver card. You can penetrate wrecks with a full cave diver card, but you cannot dive caves with a wreck diver card. Some people who only dive wrecks try to change this. But I don't see any need for myself.
The discussion to stay on intro to cave level for a while is one that will never finish. For some divers, intro to cave level will be fine, but what I see here in Europe is that most divers on that level go over their certification limits within less than 5 dives. The 1/6th rule is the first that is overtaken. Then it is the no deco and the no jump. So then I always advice to combine all cave courses together. What you see then is that people are more carefull and they have learned how to do things right. But a lot of people who take the cave diving courses in Europe are already technical divers at least on advanced nitrox level. They don't start with a single tank. I also advice people to do adv. nitrxo before doing a cave course. I never gave or will give intro to cave with a single tank.
I also don't want to see an fundies/essentials/basics of tech course as mandatory, it can be usefull for a lot of people, but not for all a needed course to progress.
I like the option to have steps between courses like from adv. nitrox the step to add some helium before going to the 60m trimix cert, but only as optional.
So conclusion about your question: if you instructor is just a cave diver and cavern instructor, he cannot bring you on a cave dive at the next level. If the instructor is at least an intro to cave instructor, he can.
If it is just a friend who wants to take you on a cave dive, I would advice not to do, but it is probably not forbidden by law.
And sometimes you get in strange situations, only single tank options, a guide that has no cave cert, but he is the only way to bring you in a cave and you are a full cave diver. Then if such a thing happens, be careful, but I know, most divers will not say no to it. But use your brains.