The datasheet you referenced is the same one I found. It is for the early metal envelope 6L6 tube. The 34 watt characteristics are for pentode mode class AB1.
Am I right in assuming that is G2 they are talking about
?
No, G1 will start to draw current, and consume power if driven positive. According to the same data sheet, page 89, you need 0.35 watt to drive the grids. This is no big deal today....we have mosfets and tubes good enough to make a decent cathode follower. In 1939 you needed a good driver transformer, and a small power amplifier to drive it. Including transformer losses, about 1 watt was needed.
G2 will always consume some power in any NORMAL pentode operation. Yes, I know it is possible for some tubes, especially RF tetrodes to actually put power OUT the screen grid due to it's negative resistance region, but this should not happen in a normal audio amp.
would it sound any different?
It will sound pretty much the same as a modern 6L6GC running the same power in AB1. The basic construction of the 6L6 hasn't changed. It got glass with the 6L6G, straight sided glass with the 6L6GA, better materials with the 6L6GB, and got a bigger plate with the 6L6GC, but inside the plate, it's the same tube.
The original 6L6 had lower ratings than the modern 6L6GC and 34 watts was about all you were supposed to squeeze out of them in AB1. You could get up to 60 watts in AB2 which does require some driving power to push the grid positive.
The 6L6 types have a reputation in some circles for being harsh sounding. Other people like them. I find them OK in a guitar amp, but they have never been my favorite tube. They can be tamed by local negative feedback, and the RCA engineer named Schade wrote a dissertation on this when the tube was introduced. A modern 6L6GC can make about 30 watts in push pull triode connection, and does sound a bit different than the same tube in pentode mode. I have always found that a 6550 or KT88 betters the 6L6GC in most every way, so that's what I tend to use.
that's what the OP was referring to. Those must've gotten pretty hot in operation.
The old metal tubes did a fairly good job of transferring the internal heat out of the tube. That's why they get so hot on the outside.
I went to a technical high school in 1967 - 1970. We learned electronics on vacuum tubes because that is what we had. Much of our lab equipment and parts were donated by the local Air Force base, including several hundred brand new RCA metal 6L6 tubes. I personally blew up maybe 25 of these, and extracted somewhere around 200 watts out of 4 of them. Did they get HOT, well the paint peeled off! I don't know how long they lasted. They were still kicking when I graduated leaving the mother of all guitar amps.....and a rather large Tesla coil behind.
We cut several metal 6L6's open after I fried them, and they had less internal damage inside them that some similar glass tubes I blew up.