I've always preferred playing non-Federal-agent PCs in Delta Green (this was 'old', CoC-rules DG). My first character (now sadly deceased) was a Berkeley professor specializing in cults (and who had briefly been a member of a Mythos cult in his youth) who was recruited as a Friendly. I think he became a full-fledged DG cell member after his fourth Night at the Opera, and died on the fifth. I just preferred playing a Friendly because I found it too challenging to play the kind of 'skilled professional' that a federal agent would have to be - I was no good at convincingly roleplaying someone who could question suspects, throw his weight around in the police station, etc. I've seen it suggested (I can't remember where) that Delta Green might have all-Friendly cells (maybe named Alpha, Gamma, Delta etc rather than A-cell, B-cell etc). I also think that (if you're talking 'unofficial' DG) it makes more sense to have Friendlies play a lead role - realistically, it would be hard for federal agents to get away with all the things a typical DG agent does - and have federal agents mostly as the 'friends in high places' who provide leads, spring you from jail etc. Of course, a lot of DG scenarios hinge on PCs using their federal agent status to gain access to crime scenes, so maybe it's best to have at least one PC be a fed...but still, I think Friendlies can make really good PCs in Delta Green.
And yes, it does sound like you easily could run a non-DG campaign using the DG rules, although some of the rules may not be applicable to all campaign styles. I've often daydreamed about writing a scenario where the PCs are a Provisional IRA unit in the late 70s - DG sounds like it would be a good fit, as you could have pregenerated characters who were hardened to violence or imprisonment.