The real reason to do 1x11 or 12, the reason it so quickly gained popularity and SRAM beat out Shimano for a few years, is that front shifting sucks. There are other reasons too that have to do with rear suspension but that's not relevant to conversions. The premium 1x systems are lighter but they depend on very clever and expensive cassettes to get there, and you need a different rear hub. In the "refurbishing old MTB" price range, the cassettes weigh well north of a pound, pretty much eating any weight savings, and you need the right chain ring too so you are looking at approaching $200 range to convert if you buy all new parts. I'm not saying no! I put 1x on my old tandem and it makes me very happy. But not for weight.
If you really want weight savings from the drivetrain, 1x1. If you can tolerate a pretty short gear range for your 1x, I think you can still get XT 10 speed 11-36 cassettes and Zee shifty bits, but I think the lightweight 9 speed cassettes are gone
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Genesis 49:16-17