I'm sorry to burst your bubble but that's not how it works. If a firm contacts you and they specifically want to get you set up for a phone interview, then that's interest. But what you're describing is basically spray-and-pray email campaigns from recruiters. Amazon does the same thing. Don't waste your time.
By the way, if they want to set you up with an "online programming test", that's also complete B.S. in my opinion. But that's just me.
"they just keep failing to reply to my messages via email and LinkedIn"
One follow-up email, maybe two max. Then move on.
My last bit of advice? Don't put any company up on a pedestal.
Edit: To soften my response a bit: don't take the recruiting process personally (nor what HNers say either!). Lots of companies do this now, and internal recruiting departments within companies have become almost as obnoxious as agency recruiters these days. Be savvy with how much energy you expend interacting with recruiters.