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.

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.