Unfortunately it has not been made clear why special (or “right”) initial conditions would be required; most scientists think that this is NOT necessary and it would most likely be unphysical. What has been established, however, is that independent of initial conditions the release of the inflation would lead to the homogeneity and isotropy we observe today. Finally, we actually observe the “remnant” of inflation by the accelerated expansion of the universe – which is the same process running at a lower energy level (which is where the “dilemma” of a mismatch of 10^120 orders of magnitude between certain energy levels comes in). Given this “mismatch”, any low probability of a theory is not immediately ruled out, it’s just more or less (un-)likely; nothing more, nothing less – and it certainly doesn’t justify to misname a theory a “myth”.

Not clear to me from the video where the 10^10^100 come from. And why an initial velocity is such a big problem. Would it overshoot the minimum of the potential, oscillate too much for too long?

The energy that is compressed in a tiny ball which exploded in a minuscule of time frame and still expanding after 13.5 billion years, equals a myth only. Why can’t we have a re-look at the Oscillating Universe and a Steady State Theory of Fred Hoyle and Jayant Narlikar. The Hindu Scriptures like Surya Sidhantha, abhorred by Science incidentally, points to one such mechanism of birth, death and rebirth of epochs of 4 YUGAS, called 1 Yuga cycle which is 4.32 million years. These Yuga cycles repeat many times (72 million times) and last for many trillion years(311.04). A food for thought.
https://en.wikipedia.org/wiki/Yuga_Cycle

We are then back in the world of statistical confidence intervals, and can reject the joint proposition of inflation as consistent with currently known theory assumptions, with 100% confidence. However we cannot have 100% confidence that these underlying theory assumptions will themselves remain robust over time.

I don’t have a problem with this. If you literally have eternity to wait then it doesn’t matter how special the initial conditions are – maybe only patches of spacetime where these conditions are met undergo inflation, hence only these patches create a universe, and if the probability is non-zero, they will eventually happen – nobody is out there keeping a timesheet of when they happen. As easy as that.

I agree, if probability is non-zero and time-frame sufficiently large, all events will eventually occur.

This is not very unusual especially when we looking for a very deep theory. In my opinion now we should have to look further with a new idea…

As it has been said – with infinite time being available all possibilities become probabilities. Nevertheless William of Ockham (Ockham’s Razor) and many others back in the 13th century had something to say about probabilities.

I find that Elizabeth’s answer is sufficient to answer this.