Interesting question... I've ticked red kite even though they were almost definitely Harewood birds, but I definitely see your point.
But where do you draw the line? Do you tick mandarin? Little owl? Pheasant?
Personally? Yes, yes and yes. In certain places and situations, but then I'm neither a serious lister/twitcher or the owner of a particularly large list (my UK life list is still under 200...)
For me, if there's an established population, living wild, for a decent amount of time then they can be counted. That can be said for all three birds you mentioned, as well as dozens of others. Even the pheasant, with a heavily subsidised population has been here since roman times and if left well alone would do fine for itself. Red-Legged Partridge? Sure, tick that too, I guess. It only begins to get dodgy when you tick it somewhere you KNOW the birds have been reared and released.
As for ticking reintroductions, the same arguments apply. It seems slightly mean spirited to not tick a bird cos 3 generations ago a boatload were shipped over from somewhere in eastern Europe and set free. Surely half the point of reintroducing once native species of any animal type is to put back what we once had for the enjoyment of everyone here? Having said that, it'd be stupid to tuen up at the site of a corncrake release, wait for it to be set free and tick it as "wild" the second it's free of the enclosure...
Besides, once you start knocking reintroductions from your list cos they're helped too much by humans, you've got to start to consider knocking off birds who are heavily and unnaturally protected (round the clock nest guarding, provision of special nest sites etc), birds who have spread naturally but only as a result of human interferance (climate change, potentially), or birds who would have, by now, been extinct but for concerted human efforts.
Hmm, longer post than I intended. At the end of the day, tick what you feel comfortable with, and what you can justify. Noone else can really tell you what not to tick. Maybe I'm just trying to justify my own "loose" listing ethics...