What incentives do I have to run Maven over Exarch/Eater?

Exarch/Eater:

1. More inherent density
2. Altars give more and semi-targeted rewards
3. 14 maps between boss encounters
4. Encounters guaranteed
5. No specific maps needed to progress

Maven:

1. Less density
2. No altars
3. 50-100 maps between boss encounters
4. Splinter-based mechanic means rolling invitations to maybe get a chance at more splinters and fewer maps between encounters (also the number of splinters dropped seems to have been nerfed which doesn't make sense)
5. Ping-ponging between 10 different maps to run an invitation
6. (SSF only) Lucking into the right invitations dropping to get a chance at more splinters
7. Maybe get a chance at an awakened gem? Maybe?

There's only one extremely small incentive to run Maven. The solution is to add packs and altars to Maven-witnessed maps while allowing people to fight duplicate bosses in invitations. Also, add alt-quality gems either via altars or as Maven rewards.

Just do 4 ways

There are no incentives to do that.

There's no reason to run maven on regular maps.
But there is on boss maps.

Conqueror, shaper and elder maps are best ran with Maven. Only 4 maps per invitation and the invitation itself is like 3 times more juicy than standard 10-map invitation.

With Shaper and Conqueror maps you also get regular witnessing. Elder doesn't because the Elder Guardian kills the map boss.

The standard map boss witness is a noob trap. Absolutely not worth the effort. I think that's in desperate need of buffing, because yeah, 50-100 maps is very unrealistic compared to just simply running eldritch altars.

However, she's terrific for Elder/Shaper/Conqueror witnessing. Elder is the easiest of the lot if you're willing to tolerate the time consumed with the Elder killing and eating bosses, spawning the guardian, and moving them to different arenas during the fight. You could even fight Elder/Shaper/Sirus with their boss fragment drops you'll be getting to gain even more loot on top. You only need to run 8 maps total to fight Maven this way which is FAR shorter than Eater or Exarch.

However, you mentioned SSF earlier. If you play SSF, the only reliable source of witnessing you've got is standard maps, so Maven is terrible if you play on that masochist mode.

3.20 Patch Notes:

Atlas Changes
Maven-witnessed Map Bosses in Tier 14 Maps or higher now have a chance to drop an Awakened Gem (inclusive of the Awakened Gems that are typically exclusive to Maven herself).

Beside the fact it's not clear, at least to me, if "Maven-Witnessed" means Maven-influenced Maps or, Innocence forbids, a map with Maven influence for which I already have its boss witnessed (Not quite sure, but I've noticed that Maven's influence does not actually toggles off now if you run a non-special invitation map already witnessed), it appears that GGG believes that "a chance" to drop an Awakened Gem for each boss you kill compensate for the lack of Altar and Monster density. Deterministically sure Altars and Monster density, I add.

Guess we'll have to wait for 3.21 to raise the offer...