Okay, let’s continue on. If trans and nonbinary people didn’t exist, you could, for the most part, look at someone and assume their gender; however, because trans and nonbinary people exist, you can’t look at someone and assume their gender. We display pronouns so that people don’t have to assume, and trans and nonbinary people can be reco…
I'm a nonbinary trans person and appreciate your allyship, but can't agree with this statement as phrased. Cis people with gender presentations that are more typical of the "opposite" sex are frequently misgendered, especially in conservative communities. This is especially true of "butch" Black women, who fall victim to gender policing alongside their trans sisters in restrooms, sports, and other areas.
In short, even if gender and sex were strictly binary and intersex people didn't exist either, you wouldn't be able to make an accurate assessment of someone without inspecting their genitals. Which, if it needs to be said, no one other than that person's doctor, lover, or caregiver ought to be doing.
Regardless, all this is irrelevant when it comes to putting pronouns in an email signature. You're not going to be looking at that person there unless their photo is also in that signature. Most people make gender assumptions based on names, but as you point out, names can be gender-ambiguous.
Gender ambiguity doesn't apply only to names like "Pat", which could be short for "Patrick" or "Patricia". It also applies to names from languages that are unfamiliar to the email recipient. While most US-Americans assume that "Julie" is a woman's name, for example, the same might not be true for someone who does not speak English as a native language. And the reverse is true when native English speakers try to determine the gender of a person with a "foreign" looking name, without any visual or aural clues. Having their pronouns listed helps resolve this ambiguity.
In any case, for a cis person, announcing one's pronouns should be seen as a simple, small act of allyship, not an onerous or oppressive burden. Some trans and non-binary people have good reasons not to publish their pronouns, including safety concerns, having multiple sets of pronouns, or wanting to be addressed without any pronouns at all. So I don't think announcing one's pronouns should be mandatory for anyone. But I do think that normalizing their publication is a good thing overall.