Re: Valve seat halfway out...
Follow-up: Applied a very small amount of Seal Lock under the seat using a fine paint brush, and then drove the seat in as best I could (appropriately sized socket). Then I applied a very small bead of Seal Lock around the top, most of which was sucked in by capillary action. Wiped off the excess. Then I carefully staked it the way the B&S repair manual shows for heads with removable seats. Lapped and reassembled.
Engine started. Ran it for an hour no-load so I am pretty hopeful I will get more use out of it.