Huge choice isn't an advantage you're offering, it's a duty you're expecting them to perform. I know, it sounds odd, but for many, choice is very hard. Also, expecting them to use a tool they've never used before to make that choice (worse if they have to book it there, and then synch it with their calendar) is more work.
I know, that's not what you mean to do; you're trying to make it as flexible as possible for the expert to fit you into their schedule. But things frequently work better when you narrow things down to a reasonable size for choice.
I got a call yesterday and needed to set up a followup. Now, it was live, so the back-and-forth was easier, but it went:
- I do need some time to think before, sometime two weeks from now?
- Let's look at next week. What times are good for you?
- After 1500 I can usually be free.
- I can do that. What day?
- Tuesday or Thursday?
- Okay Tuesday at 1500.
In email, it would probably go:
- Can you suggest a few times that will work for you?
- Sure - 1500-1700, TRF
- okay, 1500 Tuesday.
So, when you get to the "we can talk, let's set a time" part mention three or four specific timeslots that would be "good for you" first. "Or, if those do not work for you, feel free to offer a slot open on my calendar [<Calendly link>]."
Part of offering timeslots is that the expert gets a feel for how much time they have to put aside for it, and when you "feel available". But a lot of it is just you doing some of the work in setting up the actual meeting rather than leaving it all up to them.
In response to the comment: when working worldwide, it's even more important to offer times that will work for you (in their normal working hours). It sounds like you're willing/required to be very flexible, but you still want to have booked-off hours for sleep. Note - of course the "times offered" won't be the same for each person; your job is to tailor them to the expert in Australia differently from the one in Guadalajara.
Sure the response will take a week (but booking on your calendar will take the same week - and that's if they don't say "have to learn new tool? it's not that important") but your response to their response will be immediate and hopefully be a fixed time.
The key to avoiding "a month back and forth" is that one of (their reply or your response to that reply) should be a fixed slot 90+% of the time - and that is totally under your control. Make it easy for them to give you a time that you can lock in, and then commit to locking it in.
Another pattern you could use, if you think it will work better, is "Okay, my usual hours in your timezone are [...]; can you suggest some options that would work for you this week or next and I will pick one? It will take about 15 minutes." It will still take a week for response - but then again, so would anything else - but your response and book will be immediate, again, and I doubt "all your suggestions are already booked" will be common. But I still prefer "offer suggestions in their time zone, give them an easy way to set an alternative if necessary." It's easier for people to say "no, your suggestions don't work, but this will" than it is to say "okay, out of my 9-11 mornings, when am I not going to be swamped with ... next week?", and the critical part of the process is to get them to make a decision, not get them to make an appointment.