Just tried https://fred-wang.github.io/MathFonts/mozilla_mathml_test/ in Canary with enable-experimental-web-platform-features. As they say, the stretchy characters (e.g. think \left, \middle, \right in LaTeX) all have wrong heights at the moment. Still, that's a lot of progress made.
Edit: Actually, when I select any font other than "Default fonts (local only)", the stretchy characters' heights change in response to content they embrace (though still not correct -- they tend to be taller than they should be), unlike with "Default fonts (local only)" where the characters won't stretch at all. I wonder why this is the case.
Why is the goal to make it look exactly the same as TeX? TeX is great but it's rules are based on Knuth studying existing typesetting and I'm sure a lot of rules are fairly arbitrary. When I look at this in Firefox it looks finished. Brace heights don't change the meaning of the mathematics. It's a purely stylistic choice.
Knuth didn’t arbitrarily make stylistic choices, he followed long established conventions in mathematical handwriting and printing where possible. IIRC he explained this at the beginning of The TeXbook. In this specific case though, it’s not hard to realize that parentheses, braces, radical symbols, etc. two ems taller than necessary both waste a lot of vertical space and look like crap.
Edit: Actually, when I select any font other than "Default fonts (local only)", the stretchy characters' heights change in response to content they embrace (though still not correct -- they tend to be taller than they should be), unlike with "Default fonts (local only)" where the characters won't stretch at all. I wonder why this is the case.