There are people here already salty about the patch to the menagerie chest cheese, talking about how it made it worthwhile by greatly reducing the grind. I only ever did the cheese twice, and that was to grind out high power level rewards in my limited schedule. That said, I see why people liked the cheese because they likely "wined" about how long the activity is to do normally.
--BANG! "Guardian Down for Bad Pun"--
Annnnywho... I thought I'd share a possible solution to make the Menagerie worth doing while satisfying the reduction in grinding:
Make it where the Menagerie no longer needs to use the chalice for rewards and instead drops chalice tokens!
--BANG! "Guardian Down for Bad Idea"--
Now hold your sparrows, folks! Hear me out! It won't be like the old system where 20 tokens got you ONE piece of gear. Instead, you can activate the chalice for every token you get from the Menagerie, and each run gives at least 1 with more earned for playing well. You'll still need runes to select what you want, but those drop commonly enough to not be an issue; you could even use extra runes to guarantee at most one perk you absolutely want.
As for the chalice... you know how it sits in the Pursuit tab of the Directory? That thing that takes 10 years to access from the lag? Old news! Instead, let's drop it into the Tribute Hall to become a tribute as well as our new crafting station! What better place to test newly-made wares than our unofficial shooting range?
TL;DR Make the chalice a tribute in the Tribute Hall that acts as a crafting station and use the Menagerie to earn tokens that let you use it as often as you want so long as you have tokens.