From someone who speaks publically on a weekly basis, getting this impression from what Rivera says in a public setting versus what he says in a more family oriented locker room setting is ridiculous. Public speaking requires effort with a focus on maintaining an array of certain rules that some people stumble over-especially in the business and position he's in.
To assume that he does that in a more comfortable environment during his day job among close peers while doing something he loves just because he does it in a public, professional setting is silly.
Bingo. Yes today more than ever being able to speak publicly is a positive quality for coaching and definitely helps you land the head coach gig, but it has no correlation to your ability to coach. You need to remember how politician-like these media interviews with QBs and coaches are, you need to find something to say without actually saying anything. You don't call people out and you don't want to give anything away.
What matters with coaching is how much you understand about the game, how you want your team to play the game, and the ability to convey these understandings to your team. Same goes with quarterbacks. There are many smart quarterbacks who aren't what you'd consider to be particularly smart people.
Its like chess. If you take a rocket scientist and have him play chess vs a grandmaster, the grandmaster is going to whoop him, despite the rocket scientist being a smarter person while chess requires you to be "smart".