I don't think Neil effectively became an Observer through the device; to me, it seems that the device functions like a temporal tri-corder, flattening
ambient space-time to assess surrounding temporal properties (like an object/person/area's past and potential future(s)). When Neil cracked the equations, he found out how the device worked, and was thus able to use it in his quest to "become an angel", exploiting it to create the Tears of Ra compound and scanning people at the TSA checkpoint to sort out individuals with grim potential futures.
The Observers most likely use the device to assist them in their assessment of time, but they are not dependent on it to perceive time non-linearly (though it probably makes things easier). I've made a thread that outlines my ideas on how the device works in another thread (
Flattening Space-Time). You can check it out, if you are interested.

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