In principle, expecting a partner (who can drive) to share the driving 50:50 is entirely reasonable.
But that’s where reasonable ends. Your partner is a dick. I think calling him a dick is pretty generous actually. He sounds horrendous. Is he only like this about driving or does this bullying behaviour spread into other areas of your relationship too?
I can understand your nervousness around driving after a long break. I took my test at 17 and then didn’t get behind a wheel again until I was in my 30s, and that was only because my partner was having surgery (after a ski accident) so couldn’t drive and I had to take over as he lived in the sticks and was car dependent (note: I lived in central London so had never needed a car). I still remember driving home (to his) from dropping him at the hospital as it was the first time I’d ever driven a car on my own. And on unfamiliar country roads. Then going back to the hospital that evening meant driving a car on my own AND in the dark. And finally returning home was driving on my own, in the dark and the rain. I laugh about it now but it was so nerve-wracking at the time and I felt like a basket case. In the end it spurred me on to getting back into driving so like you OP, I paid for a driving lesson (in central London!) and the instructor said my driving was actually fine and I just needed experience. I started using rental cars for weekends away etc and was soon whizzing about, and these days I have my own (sports car) and am very comfortable driving anywhere (cities, town, country, hills, snow, you name it) plus have driven in countries all over the world (including Australia and Italy).
Sorry for the long history, let me get to the point I want to make. All the advice I/we can give you about building your driving confidence up becomes totally irrelevant because your partner is a dick about your driving. HE is the problem, not your driving. His attitude is bound to make you feel 100 times worse.
You don’t have many options.
You can refuse to drive if he’s in the car.
You can bin him completely.
He can learn to shut the fuck up (and that includes gasps, sighs, grabbing of door handles etc), stop mansplaining and be grateful he has a bloody chauffeur.
He can turn into a decent human being (unlikely it seems).
Your life. Your choice. You choose.
Personally I’d bin him. Rock climbing as a hobby sounds like the chauffeur driven equivalent of middle aged men in Lycra cycling as a hobby i.e. tedious for a long suffering partner and utterly selfish. But if you think there’s value in the relationship, give the ‘he shuts the fuck up’ option a try. If he opens his mouth to criticise/mansplain, stop the car immediately and give him the option of getting out there and then or shutting the fuck up. I’ve always found this to be very effective and either way, you get quiet in your car.
Edit: just to add, my Dad is an ex-rally driver/garage owner/mechanic so his driving standards are high and he makes it look easy. He’s also demanding and critical. I still remember the row about his criticism of her driving when they came home from a night out and she’d driven them home so he could have a few drinks. After that my Mum wouldn’t drive with him in the car. Ever. So years later, when I was finally driving again, I eventually had to drive with my Dad as a passenger. I can’t even begin to explain my fear. Holy hell. But actually it was fine. And now I quite enjoy driving the two of us around in my little car. I don’t know whether my Dad has learnt his lesson or if my driving is amazing (ha ha!) or if my Dad just senses that all hell will break out if he tries that critical bs on me. But it does prove there’s a way forward where everyone is happy and no-one needs to be upset.