Below is a list of areas that I think things could be improved in Battlegrounds/the surrounding experience from what I think might be fixable from the least difficult to the most difficult to implement.
I do not regularly engage in many Battlegrounds but, last week I did enough to hit rank one in each of the game modes.
Scoring in matches is a bit too rewarding to healing and not rewarding enough to other forms of support/teamwork
Burst healing specifically rewards too many points. For example, I would regularly get more points healing the person carrying the Chaosball than they would get from carrying it as every time I healed above a certain amount I got points. In Death Match, you'd sometimes see healers with low kill counts on top from all of the Critical Heals they were doing (It's a little odd as if you look up the help it's listed as just giving five points but appears to give 100). People protecting the person carrying the ball by CCing or killing the attackers don't get points. You also run into the issue that people defending in capture the relic may not end up getting many points.
The messaging around the rewards for Battlegrounds isn't very good
It's unclear the amount of rewards you'll get from doing a Battleground when you queue and the rewards only appear briefly on your screen. You can earn some cool outfits from doing Battlegrounds but, you don't see them at most of the Battleground locations (the newer spots do a better job) and you don't see much of a hint as to how you'd get them. I think you'd have more people doing Battlegrounds if they could more clearly see all that they got for doing the content. (It'd also potentially be nice to hit the button to get xp to ensure you were on the bar you want to get xp on)
People that aren't familiar with Cyrodiil don't have many options for spending the AP they earn
If you go to the Battleground areas near the cities, you can buy generic potions, generic food, an expensive motif set, materials for the motif set, and weapons that are part of several sets with AP and that's it. The Battlegrounds themselves have Elite Vendors outside but, I'm not sure most people would look at that location. AP would feel far more rewarding if we had more to easily spend it on. Maybe toss in a quest that sends people to the Golden on Weekends or add a few items to the vendor such as the tents from the various teams.
The reward gap in rewards can be a bit large and can produce frustrating/unfair results.
A significant portion of the people doing Battlegrounds are doing it for the Daily XP bonus. Currently, first and second place get around 100k xp if they haven't done that daily already and third place gets around 10k. As a result, the team in second place will frequently attempt to beat the team in third rather than try to win and the team in third will try to beat the team in second which tends to lead to matches being a blowout as the team in first has limited competition. I'd suggest switching the daily to reward just the first place team and players that reach a high enough point total during the day while also evening out the rewards between the places a bit more outside of the daily. You want third place to not be so generous with xp people afk but, you also want the rewards to be good enough to potentially keep people playing beyond the daily.
Battlegrounds are putting people in matches where they don't really want to be rather frequently and it can have a negative impact on gameplay
Many of the death matches I was put in had leavers and most of the objective matches had players ignoring the objective (In one case only myself and one person on the other team were actually doing the objectives). With five different match types in can be hours before you get in the type of match you want to play the most. I'd suggest letting players rate the match types and maps and use the rating along with some rng and adjustments for how long it's been since they played their favorite type to determine the map selection of the people playing. You can't always make everyone happy but, it'd probably be possible to make most of the game happier more often which might lead to an increase in population.