When I read your post's title, I immediately though, "5F4, 5E5, etc."
You can't have presence without a feedback loop, which will knock down rawness somewhat. The 6L6's and fixed bias will have more authority and volume than cathode biased 6V6's. It's also hard to add a more elaborate tone circuit to the 5E3 without taking away some gain (the stock control doesn't knock down the whole signal, as much as one end or the other).
That said, you can build a typical 5F4 and add a bypass for the tone circuit. Tubenit didn't mention that the other amp I always brought over to his place for jamming was a 5F4 copy. I eventually added a tonestack bypass switch in the presence pot location, because I almost never turned up the presence (my guitars tended to be bright enough for me).
The tone circuit bypass is fairly simple. There is already a 0.1uF coupling cap leading from the cathode follower into the tone circuit. We will utilize it. On the side opposite the cathode follower, we will run a wire to the bypass switch, and another from the other side of the switch to the grid of the next stage. There is a 220k resistor to ground that will be used as the ground reference for the stage after the bypass switch. See the schematic posted, knowing that it shows a closed bypass switch (i.e., a straight piece of wire).
The layout shows the points I added the 2 bits of wire, and a poorly-drawn switch. My switch was really in the presence pot hole. I'd be hesitant to run it long distances to some other location, just because there is already a fairly big loop of wire from the cathode follower to the next stage through the bypass switch. Big loops of wire increase the chance of picking up hum and buzz, as well as for creating oscillation if you add gain stages.
I'd also recommend using an adjustable fixed bias circuit, as you could probably drop in 6V6's (especially modern JJ 6V6's) if you keep an eye on dissipation.