I'm not sure "we unilaterally changed the contract without notice and you agreed because you happened to use the product that day" would hold up in any court, for any reason.

A contract requires a "meeting of the minds", which is impossible if one party doesn't even tell the other: you can't agree to a contract you didn't know you were even agreeing to.

I don't believe those terms would hold up anywhere (and this is why sites will tell you, over and over, when the terms and conditions change).