That works for some professions/working environments. I did that when I was working in an office and my entire team was in one 2x6 cubicle row. Email becomes much less important. Chat was nearly nonexistent. Now as a consultant working from home, chat is my lifeline (sometimes we even use IM while we're on the phone with each other to exclude some phone participants from a side conversation) but email is everything.

I have clients who aren't on my company's chat system and can't call me because I'm on the phone all day. Chances are if they call I can't pick up. Then there are the client's vendors who will not call me for any circumstance, because that's not their place. They're not paying me, and I hold no control over their wrk. But we'll email back and forth.

Our email system has no "urgent" flag, it's up to you to read them all as they come in and prioritize properly. If you're lucky, you'll have a PM who will spot an urgent email before you do and ping you on chat if you're a few minutes late.