The enigma machine had a serious flaw: the letters could not be encrypted as themselves. Is there an online enigma simulator I can try that does not have this flaw, which would likely make it secure?
I think you're looking for the Typex machine.
It was developed by the British and it had the major design flaw of the Enigma machine removed, which was the fixed-point-free permutation, meaning that no letter will be encrypted to itself.
A Typex simulator can be downloaded on Github (released by GCHQ). Unfortunately I could not find an online simulator, but maybe you'll have more luck.