Many people who follow the ideals of free software also value their privacy.
A service requiring you to interact with Google software in order to use it, is pretty much a surefire way to find out that you and the service provider are not on the same page about the value of your privacy or at the very least that the service provider does not understand the privacy implications, which doesn't help your privacy.
There's the odd chance that the service provider has actually researched about the specific Google software that they're using and that it's somehow actually not privacy-invading, or that the service provider has a contract with Google to protect your privacy, but chances for both of those are really low.
A service requiring you to interact with Google software in order to use it, is pretty much a surefire way to find out that you and the service provider are not on the same page about the value of your privacy or at the very least that the service provider does not understand the privacy implications, which doesn't help your privacy.
There's the odd chance that the service provider has actually researched about the specific Google software that they're using and that it's somehow actually not privacy-invading, or that the service provider has a contract with Google to protect your privacy, but chances for both of those are really low.