My goals are effectively the opposite of the tropical vacation diver. I actually got into diving with the intent to do deep cold and full penetration wreck/cave dives.
You’re not the only one who sees that in your future. That road is a tough one with lots of work to get there.
Asking my instructor proved fruitless, those three are his favorites, but I know he's not into tech stuff.
This is quite common in the recreational world. The truth is that the majority of recreational instructors know very little about technical diving and even fewer actively and regularly do technical dives.
You need to find a technical instructor you can get along with for a long term relationship. Their agency is less important as, within reason, all technical courses are very similar.
A good technical instructor would regularly dive for fun and rarely in quarries/lakes (Great Lakes excepted). They will have a lot of experience well beyond recreational levels: deep with trimix well over 60m/200ft, long decompression durations (e.g. two plus hours) and probably mine/cave diving too, all in a wide range of environments around the world. They’d have racked up hundreds of (non-teaching) technical dives with hundreds of hours spent at decompression. They will dive on a rebreather and probably be qualified to teach on more than one type.
In other words, someone who walks the walk, not talks the talk.
For your future you need to get through your introductory level diving. If PADI this means Rescue Diver is their best course and the last recreational course to complete.
You must have excellent core skills — trim (horizontal), finning (frog kick, helicopter turns, back finning) and superb buoyancy. Master those and everything else is easy. People mention GUE Fundamentals (Fundies) at this point. Do look into it.
Your future diving equipment will include doubles or sidemount because redundant gas is mandatory where you’re heading. Fundies will teach you how to use doubles, alternatively there’s Intro To Tech or similar courses.
Then, once you’re stable and competent at shutdowns, ANDP (Advanced Nitrox and Decompression Procedures) will move you beyond no-deco limits. 60 minutes at 30m/100ft plus 30ish mins at deco on a rich nitrox gas.
Thereafter who knows. Probably a rebreather as helium is so expensive.
The above takes time to skill up. A couple or three years isn’t uncommon. Longer if you need the time; it’s not a sprint, it’s more like a marathon. And it’s a lot of fun.