the invisible astronaut
"the Impossible astronaut", do you mean?
told that aliens would rip the world apart for his DNA
OTOH, this could have been a ruse (at least when there
was a ruse, if it was ever real "first time round") to conceal the trickery involved.
two humans give birth to a time-lord in the form of river after sufficient exposure to time travel
Two humans
conceived one, maybe... Only one actually gave birth.
I think that the Hitler episode effectively puts paid to there being a 'proper' new-TimeLord(/Lady), although there could have been. (i.e. something Jennie-like to have happened after the Library episode.)
It doesn't help that people like the captain have arisen, along with countless other examples of molded humans.
Molded? If you mean time-travelling, like Jack, that's... I
think... not proper Time Lord tech, the wristband. The Daleks stole timetech from the TimeLords though, back in the early days, so I can't state that for sure.
(Something I've accidentally snipped about the enemies not being efficient.)
The enemies
can be efficient, until the Doctor 'happens'. (Or the Tardis. There's sufficient evidence to suggest that
she is the subconscious Sam Beckett (or whatever his "God, Fate, or whatever" is, from Quantum Leap) of the whole shooting match, or at least sufficiently so to ensure that key things happen (Volcano Day) and, when key things do
not, that they get made to happen, regardless of the Doctor's intent ("Waters of Mars", was it?).) They are indeed apparently 'better' at besting the Doctor when allied, but even then they are imperfect. (First of all, getting their reasoning wrong. Secondly, not being able to out-manoeuvre someone who knows how to get a time-loop paradox to work the way it should. (On the whole, other Time Lords we could mention (the Monk and the Master, among them) tend to try and fight time, but the Doctor does best when he makes the flow work... Although he's said (Eighth Doctor audio episodes) that he
used to be a Meddling Monk.
So that leaves us with the question as to why humans are still alive.
It may vary, especially during the time of Gallifrey's (benign?) overseeing of universal affairs, but in Eleventh Doctor times "Basically... Run!" is the message given to anyone from the rest of the universe who thinks they might otherwise interfere. #10 tried to convey the same sort of message with the Sycorax, although whether the humans turned out to be their own worst enemy is something we could discuss.
why they've been allowed to positively teem across the galaxy in later years
That's humans for you "marvellous, wonderful humans!", to paraphrase an often gleeful Doctor (of various regenerations). He might
admire other life-forms ("oooh, you're a work of art, you are...") but it seems Humans are his absolute favourite. And although a distinct rubber-face regularity seems to be a part of the ostensibly
non-human residents, there's quite a few more Monsters Of The Week that (even prior to the reboot series, Doctors 9+) aren't strictly anthropomorphic than you find in Trek... Although admittedly quite a few
are bipedal with a recognisable body-plan and their own particular variation on a visage, and only one
major 'Monster' race really departs from that idiom. (But have done ever since episode 2 in 1963! If you ignore the Manhattan episode...)
We could talk about Red Dwarf, you know. Absolutely
no Aliens. Which might or might not explain why most non-human (never-human, like the Cat; human-inspired like various mechanoids; human-offshoots like the Gelfs). Actually, what's the Despair Squid? Originally a Terran squid but evolved, or an actual non-Terran life-form? But that's the only example I can immediately think of that
might be non-Terran in origin (and, in most of the cases that are Terran, Human-arisen as well).
But then Red Dwarf is an empty universe. Of
course there'd be no aliens in a Red Dwarf universe, and generally only humans/human-derived... because it appears there's only the one cradle of life. Compared with this, it's easier to find fault with Trek/Who/Babylon 5 or even Blake's 7 (ostensibly a human-only universe, like RD, except there's notably the race that built the Liberator). Farscape tries to do a decent job (helped by Jim Henson) but falls into the trap of rubber-facedness, augmented by rubber-bodiedness and Yoda-like implementations. The Andromadean 'Maggog' are furry-suits (could be worse, could be better), although mostly it's Human Expansion again (Nietzscheans being human origin, although Perseids appear to not be, and Trance is... well).
The
book of Battlefield Earth works with truly alien Aliens. Travolta/whoever took the concept of them and made them prothseticalised human actors though (to get him a part?). And, talking of 'Aliens', the Promethean tale explains humans by a form of panspermia (sanctioned/intended, or otherwise)
towards our race.
Although the original Alien series (while heavy on the
vaguely anthropomorphic body-plan, which also meshes with the Predator series,
especially where these two merge) seems to go by the "life is rare, and disconnected, but heads towards the two-arms-two-legs-and-a-head bodyplan" principle. At least for the space-faring races (which is humans and Predators, if discounting the prometheans), with the attack/breeding/working form of the Aliens themselves either being accidentally so similar or due to influence by their being plucked out of their original homeworld (where they were nothing special, and barely scraping by) to become the sport of the Predators/whoever. I'm not entirely familiar with the
extended Alien canon, so I may have forgotten/ignored something that was brought up in a comic series or elsewhere...
I've recommended them before, and I'll recommend them again, though, if you like books... Jack Cohen and Ian Stewart (or vice-versa) wrote "Wheelers" and "Heaven" (or vice-versa, but that way round is as good a way as any to read the largely disconnected books that probably have a non-implied timeline connection in that direction).
I think my conclusion (if I have any) is that it's easier to assume human (or humanoid) prevalence across the universe for live-action fictions, not just
lazier. Although that also extends to most animations (including CGI) too, except when it's a specific case of a "space whale" or "space amoeba" or "space wasp". And whilst the likes of Autobots are justified in copying earth-forms for their 'disguises', anthropomorphic tendencies exist in their core forms, too. When it comes to literature, you're not technically stuck with the same conventions, so long as you can convey what you mean on paper. But you have to be dedicated (c.f. Jack'n'Ian, above) to persevere with a believable non-standard bodyform rather than saying "a being taller than a human with a baby-like face" or "like a robotic space-pterodactyl". If you can get away with it by
not showing the aliens directly, or having any appearance of an alien be in the form of their choosing that is acceptable to the human observer (2001, Contact), perhaps that 'helps'. But horses (or equinoid equivalents) for courses...
(To respond to yet another ninjaing... An interesting "not-a-human human" in Doctor Who is Adric. A resident of parallel E-Space. Of course, parallel universe, parallel development, right? For a lot of pre-history human-like civilisations I actually rather assume that there's far-future humans that get adrift in time and space. Which is probably lazy of me. And I forgot to mention the Silurians, who obviously arose from the same prehistoric body-plan (thus had the potential to be humanoid), albeit that was the currently extant reptilian line, which makes it curious how so alike us they turned out given that mammals, let alone monkeys, apes and then hominids, had yet to come to prominence. Although, IIRC, originally they were three-eyed, and not either the aquaticly-adapted variant of The Sea Devils nor the lithe Homo Reptilia of the currently repeated incarnations (Hungry Earth, A Good Man..., the Dinosaurs on a Spaceship" ones and the recent Christmas thing).)
((Oh yes, and any Timelord (other than the Doc' or the Mast') that is potentially still alive was obviously on Gallifrey at the time of that Planet's Time-Locking, which means they're either still trapped, or the events of The End Of Time were such that having failed to break out of the trap they no longer exist. Of course, since Big Bang 2 and after the big time mashup that the wedding of River Song was supposed to sort out, they could now exist/not-exist/never-have-existed/whatever... hard to say.))