I agree with pretty much everything you said, except regarding Count.
Count is not good.
It can be an elite early trasher,
Actually, it can't. I mean, chapel is an elite early trasher. Ambassador. Remake possibly. Count is hardly better than steward. Of course, if you're fortunate enough to get one on turn 1-2, then it's pretty good, but this will be rare. And after you've reshuffled a second time, its value as a trasher diminishes GREATLY. You must trash your whole hand, which is a pretty big negative, and you can save one card with the top-deck, but then you have to anti-cycle it, or you can save two, but then you have to discard them. And if you want to trash everything, then you have to gain a copper. So you end up netting only 3 cards trashing, which hey, that's not bad if you can get it t1-2, but I think I'd actually take steward half the time. And steward costs $3.
or a pseudo-Mandarin in midgame (which I think is a much better card than some people realize)
Mandarin IS a much better card than some people realize - but mostly for its on-gain effect! Buying count to be a mandarin is... well, it's not the worst thing ever, but you almost always have something better to do.
and can gain you Duchies straight-up in the late game when the +$3 option won't get you to a Province.
This situation actually comes up infrequently. If you have $5+, the $3 gets you a province. So you have less than $5. But if you have $2+, then the $3 gets you a duchy anyway. So gaining it only actually gets you a cheaper card - maybe an estate. Now, that's not nothing, but it isn't much. More useful is when you can already buy a province, or when you need to do 2 duchies rather than a province, because you're behind.[/quote]
And that's not counting more unusual situations where the option to gain a Copper isn't so bad, or you really want to discard two cards (Menagerie, Tunnel).[/quote]
It has to be incredibly rare that you discard two cards for menagerie. I mean, first of all, you need to have an extra action already, then you need to have multiple copies (often 3+ or you'd top-deck) of said card, and not be able to play them. Generally we are talking copper (you shouldn't have so many of these hands in a deck where menagerie and count are important cards) or like province. So province is a situation, but I have to say, that's a really low percentage thing - that you have so many provinces that they're colliding and you need to discard them with the count to have a menagerie activate to win? Got to be SUPER rare. And tunnel, well there yes, you are going to want to discard some. But, having played count and discarded 2 cards, you then have only two cards left in hand, which is not too much to make $8 with. Of course, with the amount of gold you might have floating, you can do it sometimes. But generally, well it can't be the backbone of a strategy really, it's just too slow.
It's not the strongest card but it's so flexible, which means you can use it in a lot of different situations, which makes it pretty solid I think.
The problem is... well, there are lots of decks where if you asked me 'would you like an extra count in your deck for free' then I would take it. Though even there, often I wouldn't, and many other times, I would take it but not by much. But the problem is, that isn't the question, EVER. It's always would you rather have it than anything else you can have. And that usually means over EVERY other $5, every cheaper card, and often enough some more expensive ones. And it doesn't pass that test too often, and when it does, it tends to not come too close to winning the game on its own, ergo it's not that good.
I think it's a lot like Steward in that I've mostly used it for early trashing and then terminal money when I don't want to trash any more, it's just more complicated than Steward and not quite as good because you can't open with it every time.
Steward is a much better card. First off, the drawing on steward is generally a lot better than the money. And often enough, 2 cards is better than 3 coin, and that's not even factoring in count's drawback. Of course, count has some plusses, too, but the big deal is that count costs $5, which is a LOT more than 3, and most of all makes it a MUCH less effective trasher, especially given how it has to trash things.
Okay, so what is count's best situation? Well, it isn't bad to spike a s a trasher on turn 1 or 2, but the best thing is alt-vp slogs, where gain a copper and a duchy, or gain a copper and $3, and sometimes top-decking, are very very useful.
Okay, I don't mean to pick on you here - it was just a convenient way of running down the points I want to make about count.