Is it not dependent on GD keeping upstanding status? If, lets say, they were hacked, and had their status revoked, then the Certainly certs would stop working in some places. That sounds like dependency to me.
In the normal course of events, no. If a CA demonstrates itself to be particularly untrustworthy, the actions most root programs might take (adding restrictions, or removing it from the root store entirely) still rely on those changes getting distributed to users, and if users pick up that update to the root store, they probably picked up the update which added Certainly too.
There are conceivable scenarios where GD doing something (or not doing something) could result in Certainly search no longer validating on some subset of older browsers/clients, but they're quite obscure and unlikely. If you include those scenarios as Certainly having a dependency on GoDaddy, I feel like you would also have to say that Certainly depends on their domain name registrar to not give away their domain out from under them.
There are conceivable scenarios where GD doing something (or not doing something) could result in Certainly search no longer validating on some subset of older browsers/clients, but they're quite obscure and unlikely. If you include those scenarios as Certainly having a dependency on GoDaddy, I feel like you would also have to say that Certainly depends on their domain name registrar to not give away their domain out from under them.