Hmmmmm. Have a typo in my above comment. I meant arrow spine too stiff, not too weak. However, given the info you just provided, I'm in a quandry. Given your specs, those arrows should shoot well out of your bow.
First, be sure you're not plucking. That's releasing the arrow by pulling your hand off the string away from your face. You should draw the bow with back tension, anchor fairly tightly to your face, and your hand should come backward, not out, when you release. So, first thing is to take note of your release at every shot. If the arrow slaps off the riser when you make a good release, then the problem is somewhere else.
Next place to look is the amount of centershot of the arrow. Nowadays, with bows cut past center and skinny arrows, some arrows are pointed too far to the right for a finger release (for a right handed shooter). Look at how your arrow bisects the string.
Put an arrow on the string and rest it on the arrow rest. Hold the bow out in front of you with your bow hand and line up the string with the center of the limbs. Then look through the string and down the arrow to see where the point of your arrow is in relation to the string. The point should look like it just touches the left side of the string, i.e., the point of the arrow should be left of the string when the string is centered on the bow limbs. If it's not, build out the side plate until you achieve that look. Can do that temporarily with any kind of tape.
If it points way to the left of the string, you may need to thin the side plate a little, i.e., remove it and replace it with a piece of tape, or add point weight. See below.. Given that you're shooting skinny arrows, I think the first scenario, where the arrow may not be pointing enough to the left, is much more likely.
If it's neither of the above, you might change point weight a little. Are the arrows going where you're aiming? If they're generally grouping to the left of where you're aiming, if you're right handed, you might try adding a little point weight. If they're going to the right, reduce point weight.
Finally, you might try raising the brace height. I don't know what the recommended brace height on that recurve is. Yours 7 1/8 inches seems in the ball park, but a brace height of 7 1/2- 8 inches isn't uncommon for a relatively short, say 56-58-inch recurve.
Ok, lots of stuff to try. Only do one at a time. Good luck.