Does FAA simulator (AATD/BATD) count toward experimental rating 1,000 hours requirement? In the USA, a pilot can obtain an AEA (authorized experimental aircraft) rating to fly demilled surplus military turbojet-powered aircraft under FAA 5-1580(A).  Part of its requirements are to have logged a minimum of 1,000 hours pilot flight time, including 500 hours as pilot in charge.

A student pilot can NOT log hours in a FAA-aproved AATD or BATD simulator, unless an instructor is present.

But, AFTER someone has their private pilot's license, are they then able to log some hours in an FAA-approved AATD or BATD solo (without instructor) toward their 1,000 and/or 500 hours required for the experimental rating?  (Perhaps even using a simulation of the surplus military aircraft they want to have authorization for?)  Or, is that level of distrust from un-verifiability still there after someone has their private pilot's license?
The short answer is no. As you said in your question, the regulations you quoted require you to have logged flight time, which is defined in 14 CFR 1.1:


  Flight time means:
  
  (1) Pilot time that commences when an aircraft moves under its own
  power for the purpose of flight and ends when the aircraft comes to
  rest after landing; or


In other words, you have to go fly a real aircraft. And 1.1 defines "aircraft" clearly too, so you can't argue that it includes simulators.

But even if you could use a simulator, you would still need an instructor present. 14 CFR 61.51 talks about logging simulator time for various purposes, but in every case it also says that an "authorized instructor" must be present and endorse your logbook. That makes sense: if you're playing around by yourself in a simulator, you could be doing anything at all, including reinforcing your own bad habits. The instructor is there to ensure that what you're doing is useful and relevant, and you're learning from it.