That will work, the downside or potential downside is you are buying an entity that may or may not owe taxes, may or may not be subject to lawsuits and the asset may be depreciated already.
It can work, just be aware of some of the pitfalls.
As for the LLC's being anonymous or even remotely private, I bought my Conquest in the name of one of my LLC's, I was getting calls and emails within 5 days from brokers wanting to know if it was for sale!
One nice woman even sent me the list showing all the C425's registered in the US with owners names and if a corp or LLC, contact information for the principals.
It had my name home address, cell phone number etc, don't think the state doesn't have access to this information.
If you have no permanant residence, I would register and base it in a tax friendly state. You can try to do this just on paper, but odds are you'll get caught, especially if you are actually basing it in a high tax, high enforcement state. The casual sales tax exemption works with some states and not others, Georgia is fine with it, so is Texas, but many others don't care who you buy it from, they just want money.
I'm not 100% sure on this, but in theory, if the airplane's current owner holds it within a corporation, you can simply buy the stock of the corp.
I would think you could (as an owner) negotiate a higher selling price knowing sales tax is no longer an issue if you hold ownership this way and sell the stock rather than the plane.
Correct me if i'm wrong!