I'm going to be a little flexible and list several series that don't necessarily meet all the listed requirements, strictly speaking, but at least let me explain why. It's also true that I don't dislike nor hate everything I'm going to list, even in its current form.
Full Metal Panic - I'd say it's about time we saw a real continuation of this series. Too bad Kyoto Animation has other priorities and there's no indication that they will come back to this franchise anytime soon, which leaRAB to its (hopefully temporary) inclusion.
Mobile Suit Gundam F91 - I don't hate the movie, but it really suffered from a bad case of plot compression and should have remained a TV series. Of course, it's probably far too late for something this old to be remade.
Gundam X - The last few episodes managed to conclude the story in a reasonable manner, thankfully, but they were still rushed and less than ideal in the grand scheme of things due to its unfortunate cancellation.
Blue Comet SPT Layzner - The original broadcast ended before its time and a short OVA series tried to salvage the finale, more or less succeeding, but that still meant losing about a dozen episodes of potential development.
Giant Robo OVA - There's nothing particularly wrong with the story as it is, but the potential for sequels and prequels is so large that it's a real pity to see the project suddenly end when and where it did.
Tekkaman Blade: Missing Link - This was a really weird production, reduced to simply being a trailer for what could have been an actual OVA linking the original series and its sequel. Maybe it wouldn't be that great either, but I've always wondered about it.
Code Geass R2 - I'm actually fairly satisfied with the result, saying otherwise would make me a hypocrite in my own eyes, but I agree with the sentiment that this series should have been better handled in a nuraber of respects. A remake or even an alternative retelling wouldn't be a bad idea and isn't that unreasonable in the near-future.
Tsukihime - I'm not one of those who refuse to accept that this anime ever existed, but I can definitely see why we could use another version of the story, because some of the most interesting aspects of the original game were not included. I haven't even seen F/SN yet, but now that Type-Moon movies are all the rage....who knows? It could happen.
Phoenix - I love the 2004 anime adaptation, but it's a minor tragedy that Osamu Tezuka himself was never able to finish the original manga before his death. I know that the nature of the story makes this less of a problem than it would be in many other cases, but still....a man can always dream, can't he?
Claymore - Who knows when the manga will actually end, that's for readers to find out, but I really think the anime ended in a rather anti-climatic manner and coming up with an entirely original ending wasn't worth it.
Irresponsible Captain Tylor OVA - The TV series had a good ending, but this one stopped rather abruptly and leaving the latest conflict unresolved. Another entry that's mostly made up of wishful thinking.