The problem with Lockhart is the same problem that the PS5 has. And that is, if you're targeting a system that can fundamentally play the same titles, there's just not much savings you can get.
Let's take the APU. AMD, as tradition, charges per mm despite this not really being the primary cost driver, but it's how it works in principle. You can cut down the GPU portion to half, less than, and you'd still only cut the total APU size to 80% or so, maybe, of the Xsx one. From $122 to... $99, 23 savings, not much. Then there's storage. You can't go small, that ruins attach rate. Half the cost then for half the performance? From $100, to $50. Then there's ram, I don't even know how how well cutting it down to 8gb would go over with devs, "wtf do you mean we don't get almost anymore ram than last gen, I thought this was an UPGRADE". But, maybe another $25 saved.
In total you've gone from $500 to... $400, for a significantly less powerful machine. Maybe you save a few extra bucks here or there, but the Xsx might cost a little over $500 to make anyway so it could easily be moot. Let's face it, 20% lower cost for "half the machine" as the public would perceive doesn't have the feel of a big seller.
Only way I can see Lockhart working well is if it ditches the "plays the exact same title as Xsx" requirement AND it's a mobile device/Switch competitor.