The VPS provider has your payment records and connection logs with your real IP. Since you and friends or family are presumably the only ones using it, your identity would be really easy to unveil.
That's more like just changing your IP once, though, not putting your traffic along with hundreds(+) of others through one IP. If you build it yourself, then you have a lesser degree of anonymity.
It's really not that private either.
The VPS provider has your payment records and connection logs with your real IP. Since you and friends or family are presumably the only ones using it, your identity would be really easy to unveil.