YABU - the damage to your son's braces were directly caused by his behaviour, therefore you should have to pay for it to be fixed/replaced. The orthodontist will absorb the time/cost of one or two repairs/replacements as they acknowledge that accidents and problems can arise that are not the fault of the wearer but if the damage is caused by the wearer actively damaging/messing with it then why should they fix it indefinitely for free?
I feel the orthodontist (a) gave us completely the wrong advice of being able to wear the two halves in his mouth (b) should have somewhat apologised and (c) not be such an unpleasant person in delivering the info to my son re his braces when he never gave any guidance at the start.
A) The advice was not wrong - obviously it isn't a long term solution but many orthodontists will recommend that you keep wearing your broken retainer while they get a replacement made. Even if it isn't 100% effective it is better than not wearing one at all. As a teenager I broke my retainer in half and the orthodontist told me to keep wearing it because he could get another ordered/made within the week.
B) He had nothing to apologise for - swallowing half a retainer is not a regular occurance that he would have anticipated happening as standard with a teenager
C) Sometimes teenagers need a firm talking to because they don't understand the consequences of not wearing their braces properly. I hated wearing my retainers - I had top and bottom - because they gave me a really obvious lisp that I was extremely self conscious of at 15 so I would often "forget" to put them in. My usual orthodontist was a lovely older man who always "gently encouraged" me to wear them more, which I of course ignored until I came in to see someone who was covering him while he was on holiday and he gave me a short sharp telling off and laid out the consequences of not wearing them properly which shocked me into doing as I was told.
All of those who tell me that I am BUR, what would you do if your child was rushed to A&E due to something that went wrong that the orthodontist said they could do?
Accept that this was a random accident in the same way that when my exDH told DD she could play football in the rain, I didn't blame him for the broken ankle that she received from someone sliding on the wet grass and clattering into her - it was a possible outcome but not the "norm" and they risk assessed the situation based on the information they had at the time.