Since i played with reloading quite often (up until recently) i built it 4 or 5 times.
But only once quite early and twice only with the help of the "hunter event". Then it was really hefty. But later its near useless. (unless you have relly lots of uninhabited animal heavens + Run Fol in which case it might even later be nice. With about 2-3 free Happiness in each City)
(Without reload it should be way harder and if you then aim for the Menagerie you really should declare Nationality with those catched animals right on capture. Otherwise AI civs will capture it.)
But since summon Tiger has returned if you are really lucky and get the "hunters event" rather early (taking a gorilla is recommended then naturally if you don't have one allready) + run Fol you might be able to get it on Priesthood (and that might be quite early if you beeline FoL).
If you want any serious chance to build the menagerie you have to play with Wildlands on larger Maps with balanced Climate Zones though and really have to get Animal Husbandy quite early and catch Animals with big Stacks of Scouts instead of teching Hunters before going on the hunt. (Creation mapscript usually allows for uninhabited areas not so readily acessable for the AI of differing climates so theres quite often a place where you can find 2-3 Varieties even later.)
For Balspheraps, Svartalfar, Ljosalfar and Sidar that might be an option since they allready have some incentives to go down the recon line and / or FoL and bulb / build the Shrine with bards. (Thats when the Menagerie pays off most since you are really getting some greath bards if you get this wonder. Unless you really aim for something different.)
Might be worth too go cultural victory if one can pull that one of early. The cities should get really huge without hapiness issues that way and Cultre Wars are won easiely. (One might even try to agressively settle against an AI with high not such a huge output on Culture then.

) (if you get to the second expansion without the oposing city producing any culture you usually get the city revolting very fast. And 100 Culture might be not that far away. You do get some overlap that way but it might be worth it. Especially rather early in the Game or espacially against Barb cities. Who needs military when you can use the other civs populations envy instead.

Won't work on AI Civs on Emperor or above though i whould guess.

)
The massive culture boost from the Menagerie helps with that one as well for sure.
Maybe add the Spider Pen as a requirement but only require 5 out of 6 Cages?
Since the "Spider Mine" event happens quite frequently and "regular" Spiders really tend to survive here and there due to invisibility that should enhance its buildability quite remarkably i belive. (Not more that good on animal mastery but might still be worth it.)
Whould still be no easy feat but at least seen a bit more often...
One other advantage of FoL with that one: If you race Animal Mastery (and getting some Bards to bulb it massively speeds that one up which is very strong if done early) that way you might get some Sea-Monsters via Satyrs mesmerize which not only makes you dominate the land with your Rangers but also the early seas. Also some Elephants without battle (or you can "feed" them some Scouts in advance for XP.

) for some War Elepants.
Might not be relevant yet with the sea monsters part but maybe the AI gets to grasp Naval transport + warfare quite soon... Who knows?