Moral is fine, people are very engaged in their own project and turning out good work. Company is huge and projects have potential to make a large splash across divisions.

That's the conflict I have is: Do we all save time with more isolated and focused meetings at the cost of not know what the larger team is doing?

The overlap on current projects is non existent, but team members have occasionally partnered with other team members in the past (essentially swapping sub team members out) and there are rare occasions where someone complains about how they're blocked and one of the other sub teams has helpful advice or can pair with them.

Your last points about switching to Kanban and weekly updates is good advice, this is likely the correct answer.