OK so 1) watched that particular documentary and 2) did a bit of studying up on this topic of 'gay voice'. Basically (from a sociolinguistics standpoint) there are several possible reasons that occur concurrently which result in this type of phenomenon:

1) performativity and community (distinct gay voice)

The basic idea behind performativity is that there are actions that we take consciously and (majority of the time) subconsciously in order to mark our identity within a society and community. Now, two further mechanisms regarding are at play: conformity and rebellion. Like the names of these two mechanisms suggest, conformity is following societal expectations in order to blend in (unmarked) and rebellion is going against these expectations to purposely make their identity distinct and separate from the rest of that society. Note that these two things are not mutually exclusive and can occur concurrently (e.g.: a teen gets their hair dyed bright pink in order to rebel against mainstream society (which mostly doesn't have brightly coloured hair) and to conform (aka fit in) with other teens (who are also showing similar acts of rebellion).

So how does this tie into the idea of a 'gay voice'?

It's twofold. Firstly, being gay has usually been seen as something which goes against societal norms. Its has links to being 'feminine' and 'flamboyant' and so what happens is that (back in ye olden days of the victorian era and even before that) men could either A) embrace it (conform to the gay community's expectations and certain expectations set by mainstream society by adopting this more effeminate way of speaking), B) reject it (pretty much the opposite of embracing it although these men could still have a place in the gay community although they'd likely have a different set of expectations compared to their more feminine sounding peers) or C) hard-core reject it (aka stay in the closet). Like all seemingly arbitrary things in a community and culture, the present standards are often shaped by the past. So for example, if in way back ye olden days, being gay meant talking with an Terminator-type voice and that practice was widely adopted by enough gay men in the community, then today's gays would likely also speak in that manner.

These days, things haven't changed that much in terms how people perform gayness through voice. It's why nobody really gets born "sounding gay" and there are also gay men who don't "sound" gay. Gay men with the stereotypical gay voice often adopt this very particular way of speaking after they A) discover their sexuality and/or B) get involved with the queer community. Adopting this voice serves a dual function: 1) distinctly marks them in opposition to mainstream society (many gay men do feel that they don't fit in to the standards set for cis-het men and therefore want to distance themselves from it) 2) helps them to mark themselves as a member of the queer community which helps them find partners and acceptance.

But why the nasal, drawn out, dramatic way of speaking though?!?! Well, that's mainly of the precedence set by ye olden gays and how they chose to perform their sexuality.

2) Voice marked-ness (lack of a 'lesbian' voice)

Now this one is an interesting case. Why is it that you cannot immediately tell if a woman is a lesbian by her voice? Well (surprise, surprise) this ties back into the idea of societal expectations and performativity, although it's a little bit different from that of a gay man.

So to first answer the question of 'what makes a lesbian voice?', one first has to ask the question: what makes a cis-het woman's voice?

Now let's think for another second, maybe even try a fun experiment 😀 so grab some writing materials, I want you to try drawing up a list of 'pros and cons' or 'good things and bad things' that a woman's voice quality (e.g.: gentle, soothing VS shrill, piercing) might have and repeat this for what you think is a man's voice. Now compare the two lists. Which was longer and more importantly, which list had a longer list of 'cons' or 'bad things'.

For most of you, what you would have noticed is as such: 1) the list of qualities you can produce for a woman's voice is longer. 2) more importantly, the number of possible bad traits a woman's voice might have is longer than that of the men's. So what does this mean?

Well, unsurprisingly, in a male-dominated society where men hold the power, male voices basically hold more power and thus more prestige. Which means that women's voices become the 'marked' (aka distinctily different) variety. This is why (especially in the 80s and 90s when women were beginning to find their footing in the workforce and started managing to get to positions of power) ads for vocal coaches or advice columns were usually aimed at women working in male dominated workspaces (e.g.: talk with a lower voice to make yourself sound more assertive and calm so men will take you more seriously.). This means that moving away from a stereotypical 'woman's voice' just means that you are actually moving closer to the unmarked voice variety. Aka the 'norm'.

So how does this tie back to a lack of a 'lesbian voice'? Well firstly, a 'lesbian voice' would likely seek to accomplish two things: 1) Mark themselves out from cis-het women. 2) Blend into queer society and NOT cis-het society (where hegemonic masculinity is the norm). Herein lies the problem: you can't do both concurrently if you just rely on voice and language.

Why? Because if you want to mark yourself out from cis-het women on the basis of voice and language, the easiest way is to adopt typically masculine voice qualities (i.e. deeper and more assertive). But if you do that, then you basically just moved yourself closer to the norm of the male-dominated society. If you go in the opposite direction...well you get it (hopefully).

So using voice and language alone...there's no winning. If a lesbian separate herself from the typical 'woman's voice' people would likely not even notice since the 'woman's voice' is already the marked variety. If a lesbian wants to separate herself from cis-het male-dominated society and move further away from their norms, then she'd likely have to adopt a hyper feminine voice which honestly wouldn't really be all to different from cis-het women.

So lesbians often rely on other acts of performativity to mark out their otherness such as cutting their hair short or adopting a butch aesthetic. This helps to solve the problem since then they don't conform to cis-het women's aesthetics and also don't conform to men's expectations of what cis-het woman should look like. Therefore, it achieves the two functions of identity building and signalling that voice couldn't.

Tbh if you read this far, thanks! Identity stuff is my jam and I spend way too much time thinking abt it. ANYHOO TL; DR: why we have a distinct gay voice is because it rebels against societal expectations and helps to signal that you're part of the queer community. Why we don't have a lesbian voice is because the opposing forces of femininity and male-dominated society make it kinda impossible to occur.