I don't think anyone is saying that you work him harder at the barn as a "punishment".
IMO that is absolutely effectively what is being advised, whether it's understood as that or not. (Positive)Punishment = doing/giving something unpleasant in order to weaken a behaviour. So what people are saying is when the horse does something 'wrong' like rushing home, you apply the unpleasant stimuli (work) to home, to weaken the behaviour of rushing to get there.
BTW, I'm not saying there's no place for punishment in treating the prob, and I'm also not saying don't work the horse at home either.
I realise I 'talk in circles' sometimes
, so will use what you wrote Tiny, to try to explain my point of view further...
but if getting there only equals more work,
Aside from not wanting my horses to think of 'work' as... well... work, something unpleasant, regardless of what you do, home won't *only* equal more work... there's also the security of a familiar, hopefully safe environment, which will frequently outweigh the unpleasantness of more work.
it's making the wrong thing "hard" (not impossible, but hard) and the right thing (him just walking home, so you maybe don't work him afterward) easy.
This is such an important training principle. But another principle to keep in mind is *instant* reinforcement/punishment. Horses aren't able to associate abstract effects to causes very well, so don't understand that something that happens later is punishment for a previous behaviour.
As the horse doesn't start hurrying only as you reach home, if you let him hurry home before you attempt to 'make the wrong thing difficult', you've missed out on a lot of opportunity to do just that, and have also reinforced the behaviour by allowing it to work - he wants to go home & that's where he's allowed to go. So (complimentary to other measures to change his attitude) if when you turn for home the horse starts hurrying, I'd be turning right around again, or otherwise ensuring the behaviour didn't work for him *at the time of the behaviour*.
In addition to this, while people talk about 'making the right thing easy' they often don't focus on this or reinforcing 'good' behaviour so much. Think about what is hard about a horse going out - insecurity, leaving his home & herd, perhaps unpleasantness of carrying someone. Making that 'easy' & Good is what I'd focus on a lot more than reacting to/punishing the 'wrong' stuff.