I don't recall ever being asked whether my college had the ABET accreditation, nor seeing that requirement on a job posting, nor asking any of the people I helped to hire. OTOH, it was (and a "well known and well respected" school at that), maybe HR screens those, and ... that wasn't my job.
If you've never seen "degree from an ABET-accredited program" in the requirements of a job before, all you simply need to do is go to
USAJOBS.gov and do an advanced search for occupational series 0850 (Electrical Engineer), 0854 (Computer Engineer), 0855 (Electronics Engineer), 0861 (Aerospace Engineer), 0830 (Mechanical Engineer), etc. It's not an interview question that's typically asked; HR usually eliminates those candidates. Also, the big industry players won't be found recruiting professional engineers at non-accredited universities.
Likewise, there could be non-accredited schools that have GOOD reputations for covering material that's not part of the normal paths (there is certainly plenty of complaint about the ill-preparedness of recent graduates for any "real" work...)
Name one non-ABET professional engineering program in the US with a good reputation. The problem with the programs you've mentioned is if they do in fact cover atypical material, 9 times out of 10 it's way too application specific (in a world where technology is constantly changing), and they'll likely fail to cover fundamentals in depth. This pedagogical paradigm is characteristic of technical/trade schools, not professional engineering programs.
One major factor that distinguishes an ABET-accredited program is lab requirements. In fact, this criteria was so important that the courses which had required labs made it their prerogative to highlight the ridiculous amount of time that students would have to spend outside of lecture preparing for the 3 hrs/week lab sessions; my experience was 10-12 hrs preparation for the simpler labs. Non-accredited schools have slim to none as a consequence of either budgetary constraints or pure online coursework. So the question becomes, "If graduates of ABET-accredited programs are deemed ill-prepared for 'real' work, then what does that say about graduates of non-accredited programs?" There's quite a bit of separation amongst ABET-accredited programs in the "above and beyond minimum requirements" category as it is without bothering to compare programs that fail to overcome the lowest bar.
Another is academic rigor. You hear it all the time on this forum: I want to be an engineer but suck at math and don't want to learn calculus. Well, ABET standards do a fairly decent job of playing gate keeper, allowing only those with talent and/or willingness to work their ass off to pass muster. If the students just wants a glorified piece of paper, there are plenty of proclaimed "institutes of higher learning" that will gladly take your money to grant this desire. But if the student wants a real engineering education with industry value in the US, then an ABET-accredited program is really the only way to go.