Here's the increase I use when making these (by the way, for future reference, this booklet also had a fantastic standard mitten pattern for various sized hands, which I believe the vintage site has also reproduced, and I also use this increase in the thumb gore for those): Instead of making a stitch in a stitch, I do what Amy calls the "Make 1 Left" or "Make 1 Front" in her videos, here. It is almost invisible, very flat, looks like it's part of the original knitting you're doing, no holes. In short, you lift the "bar" between 2 sts (2 stitches which are on the needles) onto the needle, and then knit into the back of it, so the bar gets twisted or crossed. If you look at the video for it, it will make sense.
So here's how that works in the pattern-- when it tells you to Knit 1 into the first stitch, instead, knit the first stitch, then make 1 from the bar inbetween that stitch and the next one. Then knit the number of sts inbetween the increases, the way it tells you to. THEN, for the 2nd increase on that row, I do it in reverse; I make 1 from the bar, and THEN knit the next st. It just keeps it very even and symetrical. So it works out like this (let's say you're on the 2nd inc round, so you have 3 sts inbetween the increase sts):
K1 M1 K3 M1 K1. You can section off the thumb gore to make keeping track of this easier.