If you can't find an affordable phone with DP-Alt mode, you can get it working by getting clever.

Any Android phone with a USB port can have a dock attached with ethernet, a keyboard, and a mouse. Connect a Chromecast to any HDMI display. Cast to that display.

Then install 1) a taskbar app (there are dozens on Google Play), 2) enable freeform windows in the device and 3) cast your phone to your Chromecast.

Alternatively, even the shitty phones with just USB 2 dongles can enable their desktop mode by using DisplayLink; no DP-Alt mode necessary. Worse on the battery, but works over USB micro if need be.

The biggest hurdle is software support. For getting the display to work, there are plenty of workarounds possible.