f the person is actually fine, but the crew thought otherwise, and they miss important events etc
That's the thing: it's not just the flights, it's everything else: whether it's work, conferences, family events, weddings, funerals: you name it, people will miss things.
That's why they travel when they probably shouldn't (or aren't certain), and that's why it's difficult to expect cabin crew to offload passengers on suspicion. They don't have time between arriving, turning around an aircraft, and departing on say SYD-MEL-SYD to screen passengers.
TI doesn't cover the event that is missed, just the costs (after excess).
Coming back to the question implied by the thread title: should airlines rebook sick passengers for free: I think, with a medical certificate (even if it is from Dr Howlong), yes. The cost of a medical certificate, after rebates, is not free, but it is cheaper than change fees. The effort to obtain a medical certificate is non-zero. Even Dr Howlong won't repeatedly risk their ability to practice and hand them out for no reason. Airlines will notice repeat offenders, and it is within their ability to draft T&Cs to say that they won't accept medical certificates issued without physical attendance at a doctor's surgery more than once, for example, for serial offenders.
In other words, it's possible - just as the office has adapted to post-covid working, by making it clear to employees that they'd prefer sick employees to WFH rather than come into the office - it is possible for airlines to make it clear to travelling public that - if they are genuinely unwell, they can delay their travel, and, for the cost of a medical certificate after medicare rebates, they can change their flights.
Not everyone will be able to do this, but some that can, will.
The airlines can make it possible yet make it clear that passengers that abuse the privilege will have it revoked.
Will they? I doubt it.
Will most passengers travel anyway? Most are travelling for a non-deferrable purpose, so probably.
But even pre-covid, I'd cancel work travel if I was unwell - because I didn't want to be in another city, another office, and a hotel room while unwell. So those that can defer travel because they are unwell, who will face less financial cost for doing so, might.