An end fed antenna requires a means of returning current to the transmitter.
Yes, and it uses the equal, but opposite-phase, self-cancelling, non-radiating currents that flow inside the feeding coax cable (coax Differential Mode currents) to do that.
Most end fed designs rely on the exterior shield of the coax acting as that return path.
No, no, no... With respect to creating a resonant, half-wavelength long "dipole" (or 1, 1.5, 2.0wl if the antenna is used on harmonics), only the first 0.07wl of the outside of the coax shield (nearest the transformer) must carry a small current (coax Common Mode current). If the feeding coax is at least 0.07wl long (~10ft for a 40m EFHW, longer if the antenna is designed to operate on 80m), that is all the "CM current on the coax shield counterpoise length" that the antenna needs to be resonant. That length is not critical; it can be longer, but it cannot be shorter.
If the feeding coax is longer than 0.07wl, put a ferrite coax CM choke (coax wound through a FT240-31 core) 10ft down the coax from the transformer, and the EFHW antenna will have a low
swr on all of its design bands. As far as the "antenna" is concerned, it now has everything it needs to operate.
Placing that CM choke at that 0.07wl spot down the coax eliminates
most of the CM current that might otherwise flow down the coax shield all the way into the chassis of the tuner/rig/power-supply/AC-house-wiring. I use the word "most" because, even though the CM choke forces the CM current to zero right where it is placed, there can be a secondary CM current standing wave between the choke and the rig.
If that happens to a degree sufficient to be troublesome, it gets there because that part of the coax shield is a conductor that is physically in the near-radiated field of the dipole, and the current gets into the coax shield by magnetic induction. This is not unique to EFHW antennas.