In my view there is no reason why someone can not accrue sufficient experience to act as an FTE without graduating from a recognised course, but this route may limit opportunities in some companies and countries (e.g. France, where FTE'ing is more heavily regulated that in the UK). Also, it would take a long time to see everything you would see on a graduate course, so your ability to work on all trials may be compromised.
What attending a graduate FTE course does (or should do), is provides a level of experience which will allow the FTE to work effectively in all areas of testing. It doesn't mean that you know the answers, but it will mean you know where to start and how to approach the problem safely - especially when teamed with a a TP and good design team.
Where I work, an FTO can be pretty much anyone involved in the trial who has completed the required medical and safety training. They may be an avionics engineer who needs to see how his equipment is functioning, or may be an instrumentation engineer who needs to ensure the recording kit is kept serviceable. The FTE is the person who is in charge of bringing it all together and co-ordinates the testing activities on the aircraft, either from the aircraft or via telemetry comms (the Captain is in charge of the aircraft, but the FTE is the person who sequences the test activities based upon the results, priorities, fuel load, c.g., etc etc. - with the information provided by the onboard experts of course).
Under AvP67 - Mil flight testing regs. An Graduate FTE is classified along with an FTO and as such they are all regulated under the same rules. Under EASA the rules are currently being developed, but there are attempts to make things a little more regulated, which may require further training to demonstrate non-graduate FTE qualification, this of course could be very costly.
I believe that QQ ETPS course now trains both Civ and Mil personnel as FTEs. I do not know of any other companies who pay to put their staff through a graduate course, most would recruit from QQ or train on the job with the help of short courses.
nodrama - you don't mention what your aspirations are.
FTEP