Would it be best for EA to sell the Mass Effect franchise?

Best for who? Fans? Maybe. Other game publishers still follow the "throw it out there and patch it later maybe" ethos to varying to degrees so perhaps. To be blunt, the public outcry over what amounts to a game franchise with a somewhat large cult following doesn't mean much and for EA, chucking the brand wouldn't be good business.  Public consensus doesn't reflect what you're asserting.

Be honest I was mostly referring to Battlefront 2. MEA might be considered a cult classic, some day.

I don’t see how this would be good for EA at all. It would in no way change public opinion of them; only shifts in practices and what their products offer consumers will, but the IP has more potential to make them much more than what they’d get out of selling it.

I don't know if they've lost interest, I believe EA said that if they bring ME back it would need to be something relevant and fresh.

To me that means this game didn't work, so we need to come up with something new that fans will be excited about, and we don't know what that is yet.

Sell to who, exactly?

No, ME needs Casey Hudson.

Microsoft is in desperate need of more platform exclusive flagship titles, particularly RPGs and really should consider licensing the rights from EA to Microsoft Game Studios who are one of very few who would have the means and the capability to do the series justice. There is absolutely nothing like Mass Effect in their portfolio, it is potentially a killer-app they could prise out of EA's hands for a rock-bottom price and the series originated on Microsoft platforms. It ticks all the boxes; there's genuine means, motive, & opportunity there for all party's involved, it would make perfect sense imo.

Except they haven't lost interest in making Mass Effect games. Even after Mass Effect Andromeda was announced to not be getting any DLC both Bioware and EA explicitedly stated they are still interested in the franchise and want to go back again someday. 

Are they interested in the franchise, or are they just interested in making scifi shooters, though?

Funny thing is, people aren't looking for relevant and fresh, most people seem to want ME1 RPG depth, ME2 side missions and characters, and ME3 combat and storyline. Perhaps if they focused on story and gameplay instead of games-as-service they would get more sales.

The benefit those smaller developers have is there isn't the same level of expectations on them. They can make a game that might sell a few thousand copies and the players are willing to accept the drawbacks of a game developed for their niche of playstyle. Looking at games like Andromeda you can always see the wishlish of things that a developer "must do" to make the game sell. If BioWare took a more mid-tier type of development it would be panned because of the negatives that are included in making that game aimed at a more niche audience.

Its the same with BioWare prior to EA, when they were making smaller more targeted games like Mass Effect people were willing to look past the negatives that the game had and play it. Such as being much shorter then any game after EA came into the picture. Repeated dialogue regardless of the choice you made, side missions that had very little randomness to it for example the UNC missions only had four bases used just different layouts inside of modular items. Compare the reaction to those UNC missions to how people reacted to BioWare reusing a handful of dungeons with different entry points in Dragon Age 2.

Another risk with mid-tier is even with a success it might not be profitable. Hellblade: Senua's Sacrifice is being toted to being the forebear of a new generation of games, but if the information AskAGameDev gave is accurate the game barely broke even after the golden three months.

Overall right now to me the biggest problem mid-tier development has is being noticed. I was really hoping when Valve was going to allow more games to be on their platform it would mean that there could be a new wave of smaller studios. I grew up in the 486 era and remember the shareware phase of development, but there isn't really a way to get noticed anymore. Too many people have the "Steam or bust" attitude for I have listened to podcasts over the years where people wouldn't even bother with a code to get a game for free because it wasn't on Steam. Now the problem is there is too much on Steam and you cannot find quality unless it becomes a massive hit, but that requires word of mouth and to be noticed first for the last two years Valve has in my eyes become a dumping ground.