I can tell you that big barns here like to run tabs, some of which run on for months, and many people expect to be able to pay by check, like cash is no longer available to be had. Individual owners here are very easy to bring around to the way of thinking of paying cash when services are rendered. Individual owners are also less likely to have it as easy to just skip town on their vet bill...much easier to leave a boarding barn than one's own property.
I am not even sure the vets in the practice I use would want to take a wad of cash.
As a small time horse owner, I usually haul my horses to the vet clinic unless its something like an euthansia which can't be done at the vet's. My vet has better facilities for handling a horse that might get unruly than I do. If its routine, like a simple castration, I just drop the horse off, unhitch the trailer and leave them both at the vet's office until he calls and says they are ready. I'd much rather haul to the vet than pay him to come out to my place. I do routine stuff like shots myself. If I know what the charges are, I bring cash when I come to pick the horse up.
I'd always expect some notice of the vet being late. However, it seems that you DID get notice. The office called to say he was at another barn on an emergency and would be two hours late... they may not have been able to tell you any sooner. My BM once had our vet out for a routine appointment, and had to call them back to the farm as they pull out of the driveway to come back for a sick horse (unrelated to the first visit). So they could have called you just as the horse got sick and vet had to stick around much longer than expected.
The vet's attitude, though, sounds like a vet I've heard some scary stories about. Those get crossed off of my list very quickly. There are too many good/considerate vets out there for that cr@p.
I would be upset if someone berated my staff for something rather than coming to me with the complaint directly.
So many reasons the staff person may have been unable to comply such as having to leave the office for a pick up or delivery last minute that was unforeseen, or being told late by the vet about the emergency.
It can be hard to hire good staff. I don't know what the OP said to the office person, but they may have had a long/bad day and taken it harshly, told the vet that, and hence his reaction. The proper course of action would have been to explain your issue to the vet when he arrived, so he could deal with his office & policy.