That is correct. The love trial in EP6 is just a huge fantasy version of the trials in EP7 (where Miss Spoiler-chan cannot decide who to love, and seeing that once Battler is out of the picture, it just boils down to a pick between who gets to live and be loved). It doesn’t really shine through until EP3 that Beato has that form of love the Chick Beatrice has for Battler (and to be fair, a lot of people doubted it because of the trick in EP3), it’s just really difficult to spot unless you’re looking for it.
You are correct in guessing the motive. It goes by quickly at the very end of EP4, but the words there outright confirm that what you believe the motive is to be the answer (and it’s implied in EP5 with Battler trying to guess her motive). She wants Battler and only Battler to find out the truth because she heavily believes in him and styled the games as a murder mystery since he’s (supposedly) an expert in them as much as she is.
I do hope you can find the EP8 manga out there that isn’t missing any chapters or pages (since a lot of the online translations are missing those). There’s a bit that explains the core of the story partway through called Confession of the Golden Witch (and the end of it also explains a lot of vague things about the end, including how the games of the series happened).