It is perfectly possible to ride a corner correctly with the horse's head flexed to the outside
Yes, it is, though not on a horse who is falling in through the inside shoulder as this one is.
Normally in a corner you will need to bend the horse using your inside leg firmly, maintain the outside rein against the neck to stop the horse falling out through the outside shoulder and GIVE with the inside rein.
I haven't mentioned the inside hand at all, but I will here. It is sometimes necessary to request an inside flexion of the poll or neck with the inside hand. However, this is easy to overdo, and I suspect the OP can solve her problem using her inside leg and outside rein, as I have written, with some finessing of the outside rein contact, ie, she may need to be firm with the inside leg and outside rein at first to bring the horse into the corner, but should soften (not drop) the contact to avoid turning the horse's head to the outside and risk the horse ducking in through her inside shoulder on the turn. Softening the contact also indicates to the horse she has done the right thing - a release of pressure, so hopefully she will learn what is wanted and the rider will need to use less pressure over time. This rider is struggling with her horse falling in, not out.
I know it sounds wrong but the technique is that used by the Spanish Riding School of Vienna - I am so lucky to have an instructor who trained there.
Indeed, I have enjoyed reading about your adventures in Tuscany. I believe riding from the inside leg to outside hand is common across most schools of thought in dressage, though I've heard it described differently by different trainers. I have tried to give a simple explanation of the idea and a plan of action that I think a returning rider will be able to implement on a riding school horse, but I think that what I'm suggesting for the walk and trot is not that dissimilar to your explanation.
For the canter, I think this rider with this horse will have to be pragmatic, and not try to insist too much on riding deeply into the corners. If the horse is unbalanced she simply won't be able to deliver and will either motorbike or break to trot. It would be lovely if every rider had a balanced, correctly schooled horse to learn on, but in my experience this is hard to find at UK riding schools. That doesn't mean that RS riders can't progress or enjoy their riding though it does make the process harder imo.
Apologies to the OP, who has probably read enough theory to last a long while, and who sounds like she is doing a very nice job in less than ideal circumstances.