A portion of Summa Theologica is below.
The reason that it is an error to apply "all that exists has a cause other than itself" to the first cause is because, obviously, then it would not be the first cause. It would have a cause and on and on, and "in efficient causes it is not possible to go on to infinity." The basic infinite regress, first cause argument.
I think your statement may be a misstatement of: "There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself…"
The first cause cannot be caused by itself or anything outside itself. It can't be caused at all; it can only be UNcaused.
From Summa Theologica:
I answer that, The existence of God can be proved in five ways…Many thanks for your posts..The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
If we look at the efficient causes as a string of events along a closed loop, we can see that it is possible to have efficient causation without the need for the uncaused first cause.
I think Aquinas' argument breaks down when he posits that there can be no infinite (circular) regress. That assumption necessitates a stop-gap fist cause which itself cannot be caused, but which violates A's premise that all that exists had to be caused.
If all that exists must have been caused by something prior, then first cause cannot exist, by definition. Aquinas unequivocally ties existence; thus, if you are not caused, you can't exist (on your own); that forces a conclusion that first cause not only doesn't but cannot exist (because nothing caused it to exist)!
FWIW