This is going to sound like a rant - please take it as free advice and worth every penny.
Don't get me wrong, the folks you hired may well be peaches and we love them. But the world has changed - Instagram scaled to ridiculous levels with 8 people and no HR department. Build out operations - you mean repeatable processes right? If they are not already scripts on a server make them so.
You don't need people who deal with the processes of scaling or building out. You need source control tools.
Everything that is repeatable is automatable. Anything being done for the first time is a human's domain.
Everything else is either scripted and so zero marginal cost or is an anchor on your profitability.
I can really recommend this book http://www.amazon.co.uk/Race-Against-The-Machine-ebook/dp/B0...
I think I worried about the trauma of copying BigCo culture - I like Treehouse's approach and hoped I could put up a warning flag.
Having said all that, good luck.
The team that manages the web app, and most of the other teams, are small in comparison, they are certainly small and effective teams that utilize automation.
I agree with your points on automation, I just wanted to point out there are still things that require human interaction and creativity to produce.