High Latency Communication with Potential Thesis Supervisor
They sound sincerely interested in working with you. There's no right answer to this; you're right that there's a balance between staying engaged and giving them polite space to reply. It's likely they will be coping with various life events, not just work; who knows what may be filling their time and attention. It's normal that people may be responsive in one period and then less so.

I recommend perhaps lengthening the time between check-ins, so 1 week for the first follow-up, and then 3 weeks, and then maybe 6; and then stop. Alternately, you could call as much as once a week, if you don't leave a message. Most of the risk of too much contact is because of making them read or respond to a message, so if there isn't one, that's less of an etiquette problem. Ultimately, there's no way to control whether they experience this input as pushy or annoying. I would also continue exploring other potential advisors.