For example, the results are absurdly sensitive to the choice of estimating your uncertainty as a defined deviation, or a defined low percentile within it, or a defined high percentile within it. They are also absurdly sensitive to the choice of estimating a mode or a mean value.

On more erratic than log-normal, well, I mean it. There are projects that can't be fit to a log-normal, they have fatter tails. I have no idea why. A log-normal is something tractable, many projects aren't even that.

I'm still not sure I follow. In the article, the uncertainty is not intended to be independently estimated. The distribution described has a single parameter, which is the median completion estimate, and that is the only parameter needed; the uncertainty is fully derived from that. The distribution is not absurdly sensitive to this; it's just the scale parameter, so it scales the entire distribution linearly, which is a property that you would expect from fundamental symmetry.

(Unless you mean that the summary of the estimate, rather than the shape of the distribution, varies a lot depending on whether you're quoting a 95% or a 99% confidence interval. But that is just the nature of long-tailed distributions).

I'm also not sure what it means to "fit" a project to a log-normal. A single project doesn't have a distribution, unless you're measuring completion time of individual tickets within that project. The entire project's completion time might be very far off of the median estimate, but as long as the distribution assigns a nonzero probability to it, it's hard to say from just one sample whether it was right or wrong.

If you are measuring the distribution of the individual tickets and they aren't distributed along a log-normal, I'd be very surprised; it would be worth seeing what distribution they do fall into to learn what state of knowledge is being captured by those estimates.