Your example for emergency care is fine and no one really disputes that
But a more accurate example would be a plumber fixing something in your home and telling you once the sealant has dried, to apply a coating of special paint and he can sell you that paint for £100 when screwfix sells the paint for £10. You'd expect some markup to all it worth his while to stock it or whatever, but not a massive markup.
Likewise if said plumber said he would pop back in a week to make sure it hasn't leaked, and the charge you another fee you'd probably not be amused, and tell him you'd ring if any problems.
It's nothing to do with the job the vet has done or their care, it's the wraparound sales model they use to get extra cash out of you at every opportunity.
Pet plans is another example really; fleeing and worming can all be done at home cheaply as can claw clipping although not everyone wants to do that granted. £15 a month is £180 a year, it's basically a product similar to packaged bank accounts where they know if they offer it to say 500 people, 50 of them will use every single free thing included in the package and they will make very little or a small loss, but 450 of them will only do the flea and worming or only do one checkup etc so at the scale they offer it at, it becomes a money spinner. They're pushed massively by these super chains because it's guaranteed income, which is why banks use packaged bank accounts, because so few people will use everything that is included - it scales up as well, 3 cats, £45 a month or £540 a year. We clip claws weekly, worm and flea treatments are a few quid a bottle so doesn't add up to a huge amount over a year, but everything is a subscription now.
A similar thing with opticians - I go to an optician for an eye test nothing more, and collect my prescription. Every time I have to do a song and dance about how I don't want to sit and speak to a sales assistant to price me up in their contact lens package or to run through frame options because I know I can get the exact same product online for far less. Boots wanted £550 off me for fully coated lenses etc, £169 online. Haven't bought glasses from a high street store since about 2012 now, and buy contacts as and when I need them.