With DD1 we said learn in an automatic, and if you ever find it's a pain not being able to drive manual, then see if you can learn that. But at least she'll already know how to drive and have a number of years experience first so will only be 'adding in' the gears.
If she finds that she really can't cope with a manual, then by all means let her go for it. As a PP pointed out, as petrol and diesel cars are phased out, it won't even be a thing with all-automatic electric cars (and, before too long, they won't even allow us humans to actually drive them ourselves).
However, if she does ever find that not being able to drive a manual car is inconvenient, I really wouldn't think about it as 'just having to learn the gears'. Aside from the fact that she'd obviously have to take a second test, which even most people who've been driving for decades would be fearful of having to do, I think it would be a very difficult thing to do.
When I did German at A-level, there was a girl in the class who had a German grandparent, so she was already very fluent in conversational German; but as she'd learned it organically by spending time with her DGP (and with her parent who spoke it too) , like most people learn their own native language, when it came to learning all of the grammar rules and polishing conversational German into accurate German, losing bad habits and understanding what was formal usage and what was colloquial/vulgar, and increasing her vocabulary, she really struggled - far more than those of us who had learnt it since school in the 'traditional' and 'logical' way without any pre-existing immersion in it. I'm guessing that suddenly trying to relearn to drive in a manual car would be much like that.
I think it's dangerous for me to spend so much of my driving thinking about pedals and gear changes, I feel like anything that lets me concentrate fully on the road would be safer.
If a manual is really not working for you, or you're happy to restrict yourself by focusing on what you feel confident in, then by all means go ahead. However, although it can feel like a mountain to have to climb at first, controlling the car is probably only about 3% of driving at the most - 97+% of it is navigating the roads and reacting appropriately to other drivers and the ever-changing conditions and circumstances. If you can master the manual controls - and want to, of course - then once you have it, you wouldn't have to spend any of your concentration on it at all - any more than you have to concentrate on remembering how to breathe or blink.
I learnt in a manual car, had an auto for my first car (bought it from a friend), then got a manual for my second car (didn't really care, it was just the best one available for my requirements and budget), which I still have (I keep them until they die!). When I replace this one, I'm thinking I'll probably get another auto, as we now have a caravan and towing is easier in an auto and can really wreck your clutch if you have one! I haven't decided (and don't need to) yet, but, if you can (and want to) persevere with a manual, it just keeps a lot more options open to you - at least for as long as we still have petrol and diesel cars around.