I think this is a good read to learn about what tabs and tabstops really are regardless of whether you agree that tabstops should be at different positions per line or with the controversial idea of using proportional fonts (this would work with monospaced fonts too).
I will open the comments with the most obvious negative. A lot of people avoid tabs altogether and only use spaces because it is difficult to adjust tabstop positions (the technical term for tab width in this context, see the article) and you want everyone to have the same viewing experience or at least be able to customize it properly if the source uses tabs. One common argument for tabs is that you can modify the tabstops, but it is difficult enough or impossible in enough tools that some people who agree this is good still prefer spaces. What Elastic Tabstops is proposing is not just relating to tabstops (which are poorly supported sometimes as just mentioned) but also proposing an entirely new way to render them. The arguments against tabs in general are only amplified because so few tools would properly display them. The article mentions this was first proposed in July of 2006. Next month that’s going to be 17 years. Still no tools support this out of the box to my knowledge. There is sadly too much collective inertia behind traditional tabstops.
All that said, I have a soft spot for this concept and would really love to see it catch on.