As you point out, "common descent is the factor to determining what language family English belongs to.". The Germanic words that exist in English have mostly been in the language and changed with the language throughout its lifetime. There are a few exceptions like modern German borrowings like Schadenfreude, and the demonstratives 'this', 'those' and 'these' which are of borrowings from Norse. However, the English vocabulary of Latin descent was borrowed into the language. Sometimes this occurred as borrowings directly from Latin. There was a period of time when borrowing from Greek and Latin was very prevalent because of the prestige associated with these languages. During the Norman occupation of England, there was also a very high level of borrowing of French words into English. I think this may be the stage that leads you to wonder why English is not considered Romance even a little bit. We have tons of borrowings from this time period. However, we have records of them being borrowed into English from French. They were not part of the language from the start.
Here's my completely non-theoretical way of thinking of it. Borrowing is a bit like modifying a car. If you start out with an Acura and start adding components from other car companies, does it cease to be an Acura? Do you start calling it a Borla because the exhaust component was made by that company?
Conventionally, I believe historical linguists would say that English has a large quantity of Romance vocabulary (and I believe other components of Romance grammar as well). However, English as a whole is still Germanic because the origin of the language is key. I hope this helped!