In fact, I think many problems people have with email boil down to response times. If there was a way to senders could designate how soon they need a response, recipients would be face fewer interuptions and deal with email in batches when it fits their schedule.
Email has the priority field, but so few use it that you can't rely on recipients being aware of it and there are no social conventions about its meaning.
I do like your idea of designating a response time, but it probably requires too much from the sender. It might be that you could train rules on the client that could do a good job of working out how quickly things should be responded to so your client could batch appropriately.
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.
The key difference to me is that Email is broken down into discrete messages by sender and subject it is easy to archive retrieve and resend (forward) those messages with additional attachments if need be. If you can do that with Chat now great.