There are some experimental JavaScript implementations, but without browser support reflowing high-quality justified text is a non-starter on the web.
Remember, TeX was written to be usable on a 1 megabyte, 10 megahertz machine, where it ran about a page a second. One of my contributions at the time was to modify the Pascal compiler on the Sail PDP10 to count cycles for every machine instruction executed by TeX over all users over a number of months, and Knuth fine-tuned the inner-loops of TeX here and there based on the results (the code that automatically inserts kerns and ligatures got the most attention, IIRC).
I'm much more sympathetic to the point that, while TeX's line-breaking algorithm can easily handle paragraphs with different line lengths for each line, it needs to know at the start what the different line lengths are. It's not clear how to generalize it to be able to handle layouts where the length of the nth line of a paragraph depends on the earlier (or later!) line breaks. Think tall floating figures which impinge on the text area of the paragraph they're in. I'm guessing that was the real impediment in using it in Web-land.
That's a fair response, but how about changing the (CSS) specs to allow better line breaking? Surely that would take less time than WebUSB, and Google or Mozilla could quickly push it through the IETF.