Approval voting appears to avoid the biggest pitfalls of FTPT, it's easy to explain (no run-off, etc) so fewer spoiled ballots (IRV results in more spoiled ballots by a large margin), and the physical ballot doesn't have to change (voter simply allowed to fill in the bubble for more than one option).
Did a candidate get a majority (>50%)?
Yes: They win, the election is over.
No: The candidate who got the least 1st round votes has their supporters 2nd choice votes added on to their supporter's second choice candidate.
Does a candidate have a majority now?
Yes: They win, the election is over.
No: Repeat with those who had the least votes in subsequent 'rounds' until there is someone who has a majority.