I wonder if the rate of change changed at any point.
It's a question that's been discussed at length. Pagel et al. have done some research on language change using Markov chains and Monte Carlo simulations. They found that words that are used frequently tend to be more resistant to change - quite considerably resistant at that. Again though, these sorts of deep-time proto-languages are to a degree on the fringes of Historical Linguistics so you have to take their findings with a pinch of salt.
Thats interesting, if the Flood happened at the start of the Younger Dryas ~13,000 kya and the Tower of Babel followed with the dispersal and confusing of tongues then we might see traces of the language spoken at the Tower.

I'm inclined to believe we shared the same culture and language whenever it was we appeared in Africa 200-300 kya and the people who migrated away saw the fastest changes. As language evolves stories are translated. I read somewhere the click languages of Africa might be our oldest surviving language, but is it possible earlier peoples like Neanderthals and Denisovans contributed words?
Yeah, this was not what I was saying at all in the above quoted portion. It's the exact opposite, in fact.
As to the latter part - if you were arguing that these stories were retained via oral tradition, and now are arguing that the stories are translated - you're no longer arguing in favor of oral tradition. The whole reason why stories like the Sanskrit Vedas can be reliably looked at as such strong indicators of original Sanskrit language and culture is because they were transmitted very literally; it's not about simply retelling the story, it's about memorizing every minute detail about the story down to its rhythm, verbiage, and structure.
While we're on the topic of inherited myths and legends, this is something that
has been actually studied within the context of historical linguistics and the Indo-European family in particular. By comparing the myths and etymologies of various Indo-European traditions (important to note here: mythology that can be genuinely attested to an IE origin, not as a result of alternative sub/superstrates like Aphrodite, Persian mythology told through the lens of Zoroaster, etc.). In this regard, the most useful texts are, for example, not the Roman myths (by and large inherited from the Greeks), but rather from their histories (e.g. Romulus and Remus).
The below is all taken from J.P. Mallory and D.Q. Adams,
The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World, chapter 25 "Comparative Mythology". Consult that for a far more thorough overview on the topic.
So the things interpreted in this approach would be:
The
Mahabharata and the Vedas
The works of Homer
The Roman "Histories" and traditional Rituals
Norse Eddas
Heroic literature of Ireland and Wales
some very scant and carefully extracted attestations of Lithuanian and Russian sources filtered through secondhand Christian accounts
Some Armenian epic literature
So from those, these are the mythological tropes which have been identified:
Deities:
A Sky God - pretty much one of only two gods with specific lexical/semantic cognates (Skt dyáus pita; Grk Zeus pater; Lat Iupater; Illyrian Dei-patrous; Hit attas Isanus; Latv Divs, Debess tevs; [spurious] Russian Stribogu)
Sky Daughter identified with the dawn (Skt duhita divah; Grk thugater Dios; Lith dievo dukte); and also Dawn goddess: (Skt Usas; Grk Eos; Lat Aurora; Lith Ausrine)
Divine Twins: Greek: Kastor and Polydeukes; Anglo-Saxon Hengist and Horsa; Welsh Manywydan. In some cases they share a sister (Grk Helenes; Welsh Branwen)
A Chief God Who fulfils a Juridical Function (1st function): (Skt Mitra; Lat: Numa Pompilius or Mucius Scaevola; ON Tyr; OIr Nuadu). Roman and Germanic traditions have this figure losing his left arm to maintain the sanctity of oaths. The Irish equivalent (Núadu) loses his left arm in battle.
A God who is in charge of the relationship between humans and the sacred order: Skt Varuna/Pandu; Lat. Romulus; Horatio Cocles; ON Oðinn; OIr Esus; Lith Velinas. Also of note: both Horatio Cocles and Oðinn are one-eyed.
A war god (2nd function): Skt. Indra/Arjuna; Lat Mars/Tullus; ON Þorr; Gaul Taranis; OIr Ogma.
A thunder god: ON Fjorgyn (mother of Þorr); Lith Perkunas; ORus Perunu
A third-function god (i.e. a god representing the third function, namely fertility): Lat Quirinus; ON Freyr; Gaul Teutates; OIr Bres
a goddess embodying the three functions (law, war, and fertility): Athena is given the epithets
polias, nike, hugiea (protectress, victory, well-being); Iuno is described as
Seispes Mater Regina (safe, mother, queen). Also of note the Judgment of Paris, where three goddesses offer Paris each one of the three functions (Hera: rulership, Athena: victory in battle, Aphrodite: the love of the most beautiful woman in the world)
a medical god: Apollo and Rudra (Skt) inflict disease with their bows. Also both are associated with rats.
A river goddess: Welsh Don, Irish Danu, (same root of names for some Rivers: Danube, Don, Dnieper, Dniester)
A sea God (spurious, draws on similarity between OIr: triath ["sea"] and Grk Triton)
Mythic tropes:
Universe created from a Giant where the flesh becomes earth, hair grass, bone stone, blood water, eyes the celestial bodies, brain clouds, breath wind, and head heavens. (e.g.: Purusa or Ymir) Also a crossover with the twin trope where the giant is one twin, and the other twin, representing man, has to kill the giant to create the world. (e.g. Ymir, Remus, Twisto, and Yama). This may (per Bruce Lincoln) be the origin of the animal sacrifice of e.g. the Roman, Norse, and Greek traditions.
Foundational War in which the 3rd function (viz. fertility) is brought into the world by force: Æsir vs Vanir (Freyr, Frya, and Njörðr agree to live with the Æsir); Rape of the Sabine women in Roman tradition. Possibly also the Trojan War. Indra tricked into letting the Asvins into the world.
Hero and Serpent: Archetypal hero slays a serpent (often three-headed): Herakles defeats Kerberos; Horatio Cocles slays three enemies.
Horse Sacrifice
King and Virgin: King's life is imperiled, but eventually saved by a virginal daughter producing heirs. e.g. Numitor's daughter Silvia Rhea birthing Romulus and Remus
Individual transgressing against the three functions of society (i.e. law, warfare, fertility): Germanic hero Starkaðr slays a king, flees in battle, and kills for money. Mallory and Adams also cite Herakles, I assume slaying his wife/child represents the 3rd, while slaying King Eurytus is the 1st
Also the Three Fold Death: Each of the three functions is represented by a different form of death, with the 1st being hanging, the 2nd burning, and the 3rd drowning or dying near water
Death represented as a journey over a river, ferried by an old man, possibly guarded by dog(s). In some variants the river causes some attachment to the mortal life (sins/memories) to be washed away. Sometimes the ruler of the dead is the sacrificed twin.
Eschatological final battle