Your "crazy" proposition is exactly the reality at many companies: the work computer is increasingly isolated from the Internet. At my last employer their game plan for employees was to move the whole web to a whitelist approach - if you want to browse the web freely, use your personal computer or personal phone.

So most of us carted around a work laptop (connected to corp WiFi) a personal laptop (on guest WiFi or tethered) a work phone and a personal phone.

In other news, you should never ever MDM enroll your personal phone with a work BYOD policy.