This season will end with Fingolfin's death; that is already decided. And thus next season will be Beren + Lúthien.
This season - season 5 - is the season of mayfly men and immortal elves. Their lives are moving on noticably different scales, which helps to drive home the point that mortality is different, that it means something to the elves that Men just...die off...while the elves are planning bigger strategies.
After this season, we won't have a problem - the timeline of the story will slow down, so we can cover a single lifespan of a mortal over the course of the season. We'll have Beren in Season 6, Túrin in season...8? Tuor in season...9? Etc.
But we do have to get through this season.
We're not talking about 'backstory'. We're not telling a mortal story at a usual pace where a single year passes over the course of a season, and it's just a question of whether or not we delve into the history that underpins the story. We're telling a story...with characters who will quickly age out and die over the course of the season. And it is certainly true that cramming the story full of every nuance and detail from the family trees is unlikely to work well.
And yes, you can do that with Mike's idea, where 'House of Bëor' is your character, and their motivation is always 'loyalty to the elves' and the individual doesn't matter. Except....I think the individual very much does matter when we're talking about Bëor, Andreth, and Barahir, because their loyalty to elves manifests in different ways! I don't want them to be interchangeable. I want the audience to know who each of those characters is, by name. I don't want the audience to think of whoever happens to be lord of the House of Bëor at the moment as 'the guy with the crown/hat.'
Now, I'm not convinced we need to completely compress the timeline and make Barahir Bëor's son or grandson. I think we can have him be the great-grandson still. We've already joked about a plausible way to do this. Finrod accidentally calls Barahir Bëor, and Barahir responds, 'That was my great-grandfather, sir!' I think we can have Rian and Morwen's genealogy. But...I do think that we can't tell the story that way. We have to have Bëor hand-off to Andreth and Andreth hand-off to Barahir.
The way to ask this has always been...how many episodes do you want Haleth to be alive? One? Two? Three? That's all we get. Andreth likely gets two episodes - young girl love story, and middle-aged conversation with Finrod (+ old lady cameo). That's the issue here. The elves get to have a season-long arc. The Men...do not. Sure, their houses do, but the individual Men? Not so much.