>Cheap cards are on life support - the low end is migrating to integrated graphics.
That would be true if desktop CPUs get the same level of iGPU that laptop CPUs get. They don't.
A couple of reasons: dGPU market is served by Nvidia/AMD duopoly, with minimal competitive pressure because of high barrier of entry (developing GPU takes lots of resources incl driver development). Said duopoly are focused on the more profitable AI market, and are slow-walking on consumer dGPU development. A consequence of that is abandonment of low-end dGPU which has the least profit margin. They can afford to do so because there's little danger of losing marketshare to an upstart, Intel's effort notwithstanding.
Second, the desktop gaming PC market is small, relative to mobile and other non-PC segments. Adding more SKUs to serve the low-end may well not be justified in regards to ROI, even if AI was not sucking up all the oxygen.
As for the Battlemage part in the title, an easy guess is that it'll start at ~$400 MSRP, on par with the competition. How much street price will deviate from MSRP will depend on (relative) performance. If B-mage performs well, price will stay high. If not, it will drop. It was the same with Alchemist. No reason to think B-mage will be any different.
Intel is retrenching to its core competency, and I'm surprised that dGPU wasn't on the chopping block. Even as it remains, making a push to win the low-end dGPU market would be a stupid business move. There are much better options for ROI.