Interview scheduling is a solved problem you have not solved
The single largest source of delay and irritation in most hiring processes.

Look at where the days go in your last ten hires. It is almost never the interviews. It is the waiting between them, and the majority of that waiting is calendars.
Fix availability first
The root cause is that interviewers do not maintain their calendars. No amount of tooling survives an interviewer whose calendar is blank while they are actually in back to back meetings.
The fix is unglamorous. Interviewers commit to two or three fixed slots per week, blocked in advance, kept whether or not they get used. Those slots are the interview capacity for the team. Everything else is exception handling.
Then let candidates self serve
Send a link with real availability rather than proposing three times by email. It removes a full day of latency per stage, and it is what candidates now expect from every other kind of appointment.
Keep a human path for anyone who needs it. Some people are interviewing while employed and want a discreet conversation about timing.
Give the candidate the whole picture up front
Who they will meet, their roles, how long, what to prepare, what to bring, how to reschedule without penalty. Include the reschedule line explicitly. It signals that you understand people have jobs and children, and it reduces the number of quiet dropouts.
Track two numbers
Hours from stage pass to interview scheduled, and interviews rescheduled by the company. The first is your latency. The second is your credibility. If your side reschedules more than a small fraction of interviews, no tooling will save you.
Do not make the recruiter the calendar
Once a team is past a handful of recruiters, dedicated coordination is cheaper than senior people arranging meetings, and it is much better for candidates because someone owns the experience end to end.
None of this is hard. It is just nobody's job, which is exactly why it stays broken.