I have not heard a compelling case of Poynting at DC that makes me think in any way that it's useful.
Take your pick:
Ian M. Sefton
Understanding Electricity and Circuits: What the Text Books Don’t Tell You(School of Physics, The University of Sydney)
Science Teachers’ Workshop 2002
Mark Heald Electric fields and charges in elementary circuits American Journal of Physics, 52 (6), June 1984
Energy Flow in Circuits with Faraday EMF American Journal of Physics, 56 (6), June 1988
Manoj K. Harbola
Energy flow from a battery to other circuit elements: Role of surface charges2010 American Association of Physics Teachers.
DOI: 10.1119/1.3456567
Igal Galilia and Elisabetta Goihbarg
Energy transfer in electrical circuits: A qualitative accountAm. J. Phys. 73 (2), February 2005
DOI: 10.1119/1.1819932
John D. Jackson
Surface charges on circuit wires and resistors play three different rolesAmerican Journal of Physics 64 (7), July 1996
Noah A. Morris, Daniel F. Styery
Visualizing Poynting vector energy flow in electric circuitsAmerican Journal of Physics 80 (6) June 2012, pages 552-554
Let's not forget Sommerfeld:
Sommerfeld
Lectures on Theoretical Physics (6 Volumes)Academic Press
the third volume about Electrodynamics
p. 125, Detailed treatment of the field of a straight wire and a coil
There you will find an exercise about an infinitely long resistive wire. Back in 1942 the role of surface charge and the direction of Poynting vector for a DC circuits was no mystery at all. Keep in mind that Sommerfeld is considering a very long resistor, but I copy here the conclusion:
https://i.postimg.cc/pV3mnZKp/screenshot-12.pngOr, if you want to fly a bit lower, Kraus
John D. Kraus
Electromagnetics 2esection 10.20 Circuit Applications of the Poynting Vector
p. 416
on p. 418, after considering a circuit with a battery (DC) and a resistors he writes:
"In Fig. 10-19aflow lines of the Poynting vector (power flow lines) are shown. It is evident that the power flow is through the empty space surrounding the circuit, the conductors of the circuit acting as guiding elements. From the circuit point of view we usually think of the power as flowing through the wires but this is an oversimplification and does not represent the actual situation."