[T]hese sleds didn't have a cross piece on the opposing side, I suppose to accommodate material deeper than the sled - maybe it's as simple as that.
If that board is missing then it would be a large contributing factor to the sled overall flatness. That board sole purpose is to help keep the jig flat. Consider the following picture where a crosscuts sleds parts are labelled.
Image from familyhandyman.com
The board I believe you are referring to is identified as a stiffener in the above diagram. Note that I included the picture because of the label and not because I am promoting that sled design or style.
The absence of that board might not be the sole contributing factor. The choice of plywood will also play a large role in this. Near all the tutorials on building these sleds call for baltic birch plywood. While more expensive than your common plywood it is revered for its dimensional stability.
A lesser point would be if these were hand-me-downs or constructed just for you. If the former, where they were stored would be important.
[T]he horizontal surface that the wood attaches too on either side of the saw blade tends to rise or lift up unevenly
I could interpret that statement to mean there is an issue with the runners underneath the sled or that your mitre slots are not perfectly clean and it is affected by left over debris.