This is a simple activity to practice "whose" that some of my JTEs have used. The only thing you need is a basket or two.
Start out by picking a student close by and ask them if you can borrow something on their desk, like a pencil case or textbook. Then ask the class "Whose pencil case is this?" They should answer "It's Yuka's pencil case" or "It's Yuka's." If they aren't familiar with "whose" and words that indicate possession, be sure to explain it to them before continuing!
Repeat this a few times with items on other students' desks. Whose pencil is this? Whose book is this? At some point, ask the student whose item you're holding and try to get them to say "It's mine."
Next, tell the students to choose one item they have with them and ask them to put them in the basket that you're carrying. It should be something they can remember if they see it (like a pen with a specific design or a well-used eraser) but not anything with their name written on it, since that would make the game too easy. If you're doing team-teaching, it helps to have two teachers collecting the items. Once all of the students have given an item, mix them up in the basket or baskets.
Ask the students to put their heads down on their desks, and redistribute one item to each student. Once every student has an item, they can put their heads back up. If any students have their own item, ask the class to put their heads down again and redistribute their item with a few other students to mix things up again.
Now that students have their own item, they need to ask their classmates "Whose ______ is this?" Their partner can answer "It's so-and-so's" or "I don't know." If it's their partner's item, their partner should answer "It's mine" and take their item back. Once the student has found the owner of the item, they can sit down.
If the students enjoy the activity, they can play another round with a different item. If the students don't want to part with their personal items, you could adjust the activity by having them use pictures that you hand out or they draw.
We used to do a variation on this with elementary and junior high where they had to find each other's pictures. Give out small scraps of paper, have the students draw a picture and then collect them, shuffle and pass them out again. This way you didn't have to worry anything happening to their stuff. For a picture we'd either get them to draw their treasure (which tended to take time) or give them a minute to draw a picture of the ALT (which is less meaningful but entertaining and faster).
Assuming that you followed the directions above then that is a ridiculous comment from the "lead JTE". If they'd have quietly took you aside and said they weren't so keen on it then I'd say fair enough. But to have a meeting with the VP about that? I could have understood concern about COVID but the lead JTE's explanation makes the school sound like a weird cult. You did nothing wrong. Just shrug, shake your head and do something else.
I hope they had that meeting with you before doing the lesson. It's never a great feeling when the teacher is 'too busy' for a proper meeting before a lesson so you can't get the okay on your material and then they get upset about something afterwards. It's one of the reasons I don't lean too heavily into video game IP. Once burned, twice shy as they say.