It is basically confusing an uncle with a grandfather, or an aunt with a grandmother.
These languages are indeed relatives, literally part of the same family, but we usually distinguish whether they are lineal ancestors or not. If you said "Julia is your grandmother" when Julia was actually the person's aunt, that person might get very confused about your claim!
People's aunts do often have a cultural effect on them, but we can still be more specific and distinguish them from their grandmothers.
No, the succession (Sanskrit, Greek, Latin, post-Latin) was given mainly in terms of the apogee - one peak follows the other in time. When I wrote «Sanskrit entered into Greek, Greek into Latin, Latin into English» I described the phenomenon, not the fact.
The literal matter (of direct inheritance) was meaningless to me in the composition, because the whole point was not and is not "what took from what": the whole point is the contribution that those languages give in the understanding of the current languages they informed.
Only, simply, non technically, that Sanskrit literature happened, then (non-committal irrelevant sketchy for-purpose-of-representation "then") Greek literature happened and you can find the Sanskrit terms modified (not literally them Sanskrit terms, more crisply the presence of the roots) into the Greek terms, and then (again) Latin literature happened, and you can find the Sanskrit and Greek terms modified into the Latin ones, etc.
Phenomenon: "one [as-if] entered into the other"; reality: "a common Language is revealed in their terminology"; the important part: "we can better define the terminology of current languages that share that property through the consideration of the languages that _substantially_ constitute their roots".
Except the rest of your arguments fall apart if your point is just "Sanskrit and English shared a common ancestor thousands of years ago and English (and some actual ancestor languages to English) pulled in some Sanskrit loan words". As other commenters mentioned, lots of languages (and languages with long and robust literary traditions) fit this bill, so from an English speaker's perspective there is nothing special about studying Sanskrit. If anything there are a host of more direct ancestors.
The perspective in your post is rationalistic and not empiricist. There was no theorematic argument intending to prove the importance of Sanskrit (and Greek, and Latin) for the understanding of English based on historical notions: it was a witnessing of the importance of Sanskrit (and Greek, and Latin) for the understanding of English based on experience.
It's just that I cannot allocate the safest amount of effort in composing. I thought it was clear enough.