In order for Dark Shadows to still be Dark Shadows, it needs the same heart. That heart is really the love of Barnabas and Josette. The character of Barnabas is really who he is because of Josette and her death. He constantly tries to re-create her; he wants to find her again desperately. If you don't have that crucial story element, you don't have Dark Shadows. No matter how many bizarre things happened during the original series, the main element that tied them all together was Barnabas and his search for his lost love (from his introduction onward): The kidnapping of Maggie, 1795, Adam, 1897 (Parts 1 and 2), the Leviathans. In 1970PT, Josette played a crucial role, because one of the main reasons Barnabas was so intrigued with this alternate world was because he learned from the book that he married Josette in that time band. Josette was even instrumental after KLS left the show. It was because of Josette that the Angelique/Barnabas story of 1840 happened. The love of Barnabas and Josette even carried over into 1841PT, even though one character was dead and the other was played by a different actress.
If you don't have the love of Barnabas and Josette, you don't have original conflict, without original conflict you don't really have an original story. You can have a sequel to Dark Shadows, but if it doesn't have the same heart, it won't be the Dark Shadows we know. No matter how hard the writers could try to carry over the same feeling and ambiance of the original, if they don't carry over Barnabas and Josette, it will be a different show, even though it shares some story elements with Dark Shadows.
Let's take Night of Dark Shadows for example. I love the film. I think it is a great new story for the Collins family. However, if Night of Dark Shadows was to be made today with completely different actors, it wouldn't really be recognized as being "Dark Shadows." It's a whole new story that takes place in the same setting as "Dark Shadows," has some of the same character names, but it's different. What made Night of Dark Shadows work in 1971 was the fact that we knew all of the actors as being part of Dark Shadows. There was a gothic atmosphere and storyline, but without the actors, it could have been "just another Horror movie." It wasn't though, because it used the actors we knew, and that is how we identify that film as being Dark Shadows. Because we can't have the same wonderful actors or the truly awesome production style of the original show today, the way you have to look at it is this: What parts of the story can you remove, change the names of the characters, and make a new story for, and still clearly recognize iit as Dark Shadows?
There are two. The first is the arrival of Victoria Winters, primarily because this was a truly original storyline. You can change all the names of these characters and toss them in a different series, while keeping the same storyline, but it would still be recognized as the story of Art Wallace, the first story of Dark Shadows: An orphan arrives in a small fishing town to become governess to the heir of a mysterious estate, a mansion filled with the secrets of those who live there. That's Dark Shadows no matter how you look at it. The second is Barnabas's search for Josette. Certainly the story of searching for and trying to re-create a lost love has been done before, but never in the fashion of Dark Shadows. Never before did a vampire have the depth Barnabas Collins had. Never before had the search for lost love been so unique. Cursed by a witch, a vampire finds himself trapped in a world 200 years from the one he knew. Yet, even though 200 years have past from the time he was chained in his coffin, he still loves one woman, the woman the same witch who cursed him took away. He vows to find her again, to make her the bride in death that he could never have in life. (While adding Frid's "reluctant vampire" to the mix, of course.) Distinctly Barnabas and Josette, clearly Dark Shadows.
Let's see what other stories there are... Dr. Hoffman trying to cure Barnabas? Nope, not original; done before in House of Dracula. Adam and Eve? Nope. Also done before by (Shelley and) Universal: Frankenstein and the Bride of Frankenstein. How about the werewolf and the search for a cure... Again, Universal did it first. The haunting of Collinwood in 1969? Turn of the Screw. My point is Dark Shadows itself relied so heavily on material that has been done before that the only elements that are uniquely Dark Shadows are the original characters, the two main original stories, and the original cast/production/music. Someone could carry over "non-Barnabas and Josette"/"non-Victoria arrival" elements of Dark Shadows, but those elements would be carried over from an earlier source. There could never be a new Dark Shadows that didn't have Barnabas and Josette, because it wouldn't be the Dark Shadows we have come to know. You can have a sequel to the series, but if it didn't have Barnabas and Josette, you could give it a new name altogether: The House Where Death Walks, Shadows on the Hill. You could even give it an old name: Frankenstein, Turn of the Screw, The Wolf Man... There can be a new series that carries over elements of Dark Shadows and not be named Dark Shadows. Likewise, a new series named Dark Shadows wouldn't necessarily be the Dark Shadows we know without the crucial elements, without Barnabas and Josette.