It's not (only) about pay, although your 28% statistic doesn't capture how much pay had been eroded by inflation since 2008.
It's playing thousands a year for professional membership, insurance and exam entry, none of which you can practice or progress without. And it the biggest thing is the lack of training posts. You say yourself "you start at the bottom and work up" - that used to be the idea. But the government isn't funding resigtrar training places and even if you manage to get one of the very few available (after sitting many rounds of exams at about £400 a go) you then need to move across the country to wherever you get allocated. Once there you can't settle either as you will be expected to rotate every six months for years.
It's incredibly disruptive to family life, you can't buy a house even if you could afford one, because you don't know where you'll be in months or years time, relationships break down, we're one of very few medical families I know who are having children and it's only because we've stopped trying to fight the system and delayed careers and opportunities significantly to prioritise other things in life.
The thing is the need for these positions hasn't disappeared, they are just being filled by internationally trained doctors who have already reached registrar or consultant level abroad, but quality can be patchy. In the UK our standards are actually very high so there are plenty of countries like Canada, NZ and Australia who happily recruit from the UK where younger doctors could go and get training and experience. That's such a false economy because the UK gov has paid extensively for their medical school education even with the 9k fees being paid back.
There's also just not a lot of respect for people wanting quite basic things such as efficient HR, on time and accurate pay, being able to take holiday entitlement even if you give months of notice - the message is always, you're a doctor, you're well paid and can put up with it.
These are the things the BMA is pushing for and was promised after the last round of strikes, the reality is the government have back tracked on their promises and are now painting a picture of doctors just wanting more, more, more when actually pay restoration, exam fees and training position was all supposed to be agreed last time.
Honestly, I'm surprised people aren't more angry. The quality of care you're receiving if you go to hospital now is getting steadily worse year on year as we export our best talent and import the cheapest.