그래요 comes from 그렇다 which is the form you will find in a dictionary.
When you negate the verb, you use the grammar pattern X-지 않다, where X is the root. So 그렇지 is actually 그렇+지.
I was wondering the same too actually. I think "my home" is perfectly fine.
Because technically 집 is the physical house, while home is isnt. English is weird.