The only place you have even the faintest chance of self-financing into is NTPS in Mojave - which is to say that you only need a major corporation behind you, rather than a major government for ETPS., NTPS do take a tiny number of graduate "minimum wage" interns who usually do 3 years there and whilst there do the MSc in Flight Test Engineering, which is worth a heck of a lot and a fantastic way into the community.
I started with a PPL and a BEng, and have migrated without ever being unemployed to CPL, PhD and a succession of rather fun jobs, mostly fairly well paid, as well as seeing a pretty large proportion of the planet at other peoples' expense, and clocking up a little over a hundred types, most of those in flight testing. Not much of that time however was ever dong a mainstream industry job - probably 6 years of the last 21 have been spent in anything that you'd ever find listed in a career guide.
There's not a hope in Hades of getting a full time test pilot job with *just* a CPL and a BEng. There is a small chance of getting a job as an FTE; once there you can make your own career. There are Test Pilots who started as the junior assistant-to-an-assistant FTE somewhere: and eventually ended up in the left hand seat.
So - in my opinion, there are some fantastic jobs to be applied for by people with a good selection of engineering degrees and pilots licences - and Test Pilot is one of those, but right at the top of the ladder. But, none of these jobs are mainstream or easy to get.