Kind of ignores how DTMF phones worked. The tones are actually two tones played simultaneously. Each row and column are separate tones. Laying them out in a circle could work but its not very practical.
Later on, the phone regulations were changed, and you could own a phone. That's when cheap pushbutton phones came out, that had a switch allowing you to choose between pulse and tone dialing, because you still had to pay more for touch-tone service and some regions didn't offer it.
The weird thing was, sometimes you had to use both, for instance at my college, the phone system used pulse dialing, but I had to enter a "calling card" code using tone dialing in order to call my parents and put the call on their bill.
There were phones out there that had a keypad but only supported pulse dialing too; if I remember correctly, my parents had one when I was a kid.