Is there any good way to explain to users that I can't fix other
organizations' websites and that if the easiest option is to use a
different browser, that's not necessarily a bad thing?

Seems I get a lot of calls lately about sites not working with IE8, or
IE7, so I say "does it work with Firefox" then when they say "Yes". I
say "okay, then". But they want it to work with IE, so I haven't
"fixed" the problem if they can't see what they want to see where they
want to see it.