what's been working for me decently is just setting the clock 20 days forward and then scanning my island for thought bubbles sometime in the afternoon. often there's already someone who wants to move out after those 20 days but if there isn't, I start TTing day by day and within the next few days (usually 1-3) there will be someone. if it's not someone I want out, I TT 20 days again and repeat the process, or if I don't want to end up too far into the future, I return to the current date and start again from there. it's pretty straightforward. it's probably not the best method out there but it does work. you can get several villagers wanting to move within half an hour, in my experience it often takes approx 5-10 mins for someone to ask.
also, in my experience it doesn't really matter if you talk to villagers or not, or who you talk to. and I haven't tested it but it might be faster to get villagers out if you TT more than 20 days at once. if someone's tried that I'd like to know~