Yaxley does think they were after Barty, because the assassin seems to have lurked there for a while, and if he merely wanted to slay Council Members he (or she, I suppose, I didn't get much of a look at the murderer) wouldn't have had to wait so long.

I've no idea what Barty was up to, but surely Yaxley does?

I'd have no fear about Glendower springing a leak to the press. I'm not sure how long they'll be able to keep this hushed up, though. Everyone who was at the Ouroboros knows.