Exactly. The electric field inside an ideal conductor is zero so the work done on the charges in the wire (E*J) is zero.
You're simply exchanging one imperfect, incomplete model for another. The electric fields inside a conductor, even a superconductor, are not zero. Your model just says they are because that mostly works out in the macro domain that it is intended for. What goes actually goes on is more complex. Quantum mechanics aside, I fail to see how you can continue to maintain the the energy flow (whatever that is) in the DC case is due to fields outside of the conductor without stating what those fields are. Poynting vectors are not fields. There is the magnetic field which is unchanging for DC current and then a static E-field where the conductor has a net charge. Neither of those can do work on charges. Now you can have a wrong or simplified model that still predicts at least some things correctly, so if you can have a battery on one end of a pair of wires and a load on the other and you have some diagram with some arrows that shows the E-field of the battery being magically transferred to the other end to do work on the load--whether it is Poynting vectors or monkeys with wheelbarrows--you still haven't explained how it came to be that the charge density is what it is at the load end of the wires.
You can include the electric field inside the wire if you wan't it won't make a huge different to the result. If you do a full E&M analysis you will find that the vast majority of the energy density is outside the wires. There is very little energy density inside the wires and very little power flow by normal formulas, such as qE or E x B.
It's certainly a bit of a matter of semantics. "Power flow" is not really an observable property. The power produced by the battery and dissipated in the resistor/lamp are easily defined but the power flow requires a bit more care.
The value of a field-centric approach is that everything is well defined locally. I can look at a tiny volume of space have some parameters defined there (E, B, Q, v), and assign that location an energy density (E^2 and B^2) and a power transport vector (E x B) without referring to the rest of the system. Then if I want the total, I just add up the local values over my chosen volume or surface.
If you try to assign the power flow to inside the wires, you can't come up with a local, consistent model of the energy transport. The energy transport I*V, but for that equation to make sense you need to define a point with zero voltage. If the "0V" point is the negative battery terminal all the power will be transferred via the positive lead and no power through the negative lead. Define the + terminal as zero volts and all the power goes through the negative wire. You can even have more than 100% of the power flow through the positive lead, but some of the power return via the negative lead.
This all doesn't make much sense, so we tend not to do it. Instead we look at the total circuit, analyze it, and say "1 watt flows from the battery to the lamp, lets just say it flows through the wires" That's all fine, but not very quantitative.