Bad Manners and Brimstone

I think you handled it fine.

However, as the admin for their boss, I would now send a request that they open their calendar for you to see their availability. That is pretty much standard protocol in my company.

Once you have access to their calendars, it should relieve you of having to do the back and forth emails before scheduling.
I much prefer people to just send me a meeting request for a time that is clear on my calendar.

Of course that presumes that I'm keeping my calendar up to date. I think meeting scheduling runs so much more smoothly if everyone does.

This is how things are usually done in my organization. Of course, things do come up, and meetings sometimes have to be rescheduled. But I much prefer a minimum of emailing. Check the calendar of everyone who needs to attend, send a meeting request, and generally, no further communication is needed beyond the invitees accepting the request.