We can finally play that Afterburner game which never got a home port (that I know of)!
The name is difficult to theorize but In terms of price, using commercial components such as CPU and GPU from PCs makes the console more expensive than doing good R&D in house.
xbox 360 is more powerful than Lindbergh but was sold for $299 due to massive subsidies.
This console would be a new Sega Saturn a bad value for money and it would only have reasonable sales in Japan but not in North America, selling above $299 is doomed, poor sales and selling at $299 Sega would enter into a price war with MS without having deep pockets.
360 also took A LOT of shortcuts to hit $299, including the cheap SKU having no internal storage (step back from OG Xbox), no built-in WiFi, no built-in HD-DVD or Blu-Ray (affected all models), less ports for external peripherals, etc. And all early 360s suffered from RROD, which while partly outside of MS's control, well MS definitely didn't mitigate the problem with how the early 360 models were assembled.
I think if SEGA made a 7th-gen system, they'd of taken an approach more like Nintendo, and just exited the power crown chase going on between Sony/SIE and Microsoft. What they would've done to differentiate themselves and not end up caught behind Nintendo's motion control push, I dunno. IMO that would've been the time for SEGA to try a different business model and try fully synergizing arcade and home efforts on all layers, not just in terms of shared architecture and software library like NAOMI & Dreamcast did.
FWIW SEGA wouldn't have had to enter a pricing war with Microsoft in order to stave 360 off; the 360 had a decent launch, but didn't begin building real momentum until Gears of War, and then finally fully took off when Halo 3 dropped. That's a period of about two years after the system first came out where, if SEGA made the right calls with software exclusives, could've probably eaten into 360's Western market share. Revamp SEGA.net to bring it in line with Xbox Live (or better), keep up the NFL & NBA 2K series, try and set something up with Epic Games (I dunno what the relationship between them and MS was for Gears, but maybe SEGA could've used one of their Western studios to co-develop an Unreal spinoff with a team from Epic), etc.
I'm not saying any of this would've helped them beat the 360 in that 2005 - 2007 period, but it might've been enough to eat up some of its Western market share. Which, well, is the main reason 360 was even able to stay pretty close to PS3 in total sales; it just dominated PS3 in the US & UK that gen. A SEGA console that gen would've eaten into Xbox's share in those territories and that ultimately would've been beneficial for Sony & SIE, I feel.