Yes, the company I’m contracting at currently has done exactly this. We’ve kept Slack around for calls and for monitoring alerts, but other than that it’s all in Basecamp.
It has made an incredible difference. As a remote worker, I always felt stressed if I didn’t respond to a message straight away as it might appear as if I’m slacking off. With Basecamp, a delay is expected, so I finish my tasks before taking a look through my Basecamp notifications. I’m vastly more productive and far less stressed.
In addition, the proposals and comments on Basecamp are in far more detail and conversation is typically better thought out.
Could not recommend it enough.