Why do these run so high price-wise? I haven't looked at the number of sets issued, so I apologize if it's low. But I see completed listings for these around $40+, up to $80.
In 1982 and 1983 the mint only sold the mint sets at the mint. Normally, you can order a mint set by mail. But in 1982 and 1983 they did something different. The only way to get those mints sets was to actually visit the mint and purchase in person.
https://www.cointalk.com/threads/1982-1983-us-mint-sets.195952/ Keep in mind this thread is a bit dated..
There is only U.S. mint Philadelphia or Denver souvenir sets for 82'-83'. The mint never issued mint sets for those years just the souvenir sets which only include P or D mint coins. So in essence you'd need one of each branch to come close to what you'd get in a true mint set. There are 3rd party put together sets for those years but they also sell at a premium.
If you want well struck coins from good dies for some of these coins like the '82-P quarter you can just about give up looking. Even the services are giving high grades to poorly struck coins. I've got one coin that might be almost a true Gem. The rest are a mirage, I believe. The souvenir sets are the pits but this is the source of many of the services' highest grade coins. If you want nice coins look in the Numis News sets. About one in ten or twelve is gemmy.
The '82 and '83 souvenir sets were available at least briefly through the mail. There was no run on them but the numismatic press failed to report their avaibility and it appeared only in the general press. It might be this failure that resulted in the return of the regular mint set in 1984. Sales of the souvenir sets were dismal.