Also, if you cut the ground trace going to the top left point you marked, the sound stops working, I think the square component above it is an audio transformer of some kind.
Maybe the resistor you're adding is putting a dummy load across the speak +/- connections. Don't know why it would need to be modified to work as expected though! Perhaps another connection elsewhere has gone bad.
Sometimes joined links can do unexpected things on the Speaks. I had a Speak & Spell that would duplicate one of the letters on the screen to a couple of spaces across. Turned out to be an accidentally bridged connection near the KEYBOARD area, unexpected, but you can test it by pressing loads of keys at once, and you'll see extra letters light up! It makes sense when you look at the schematic, but was a surprise at first. So yeah, is this Speak otherwise unmodified or is there extra solder on there that could be causing troubles?