phytoestrogen != estrogen

If this were true, all the pre-transition trans women I know would be eating a LOT more soy. Unfortunately, it's not, and can even have the opposite effect, by binding to receptors that estrogen would, preventing it from acting normally.

Not quite, because while you are talking about estrogen in beef and birth control pills (Diane 35, presumably), you are talking about phytoestrogens in soy milk & co, and specifically isoflavones.

Phytoestrogens have a similar structure to estrogen, and can thus bind to the estrogen receptors, but they have only a weak estrogenic effect (the opposite might actually happen). The effect of isoflavones, at least, has been studied [1], finding "no significant effect" on "T, SHBG, free T, or FAI" - this despite the trials featuring subjects whose "soy protein and isoflavone intake greatly exceeded typical dietary Japanese intake", which was said to range from "25 to 50 mg" in Japanese adults.

What effect isoflavones have might be debated, but that comparison remains apples to oranges. A daily dose of estrogen greater than 25 to 50 mg would definitely have an effect. Even a tenth of that would have a very noticeable effect. It doesn't, so just saying "170'000 ng per serving" tells us nothing, not when the comparison includes birth control pills.

Hell, that comparison would even remain apples to orange if you just compared animal estrogens, or those estrogens that can be found in pill form. Diane 35 contains ethinylestradiol, which is much more potent than conjugated estrogens (Premarin & co) or micronized estradiol (Estrace): while 95% of the estrogen you ingest is going to be inactivated, ethinylestradiol bypasses most of that.

Isoflavones do not, they also undergo first-pass metabolism, so the comparison is doubly flawed.

[1] https://doi.org/10.1016%2Fj.fertnstert.2009.04.038