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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?

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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.

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    $\begingroup$ It also had more irregular stepping, which made it better. $\endgroup$ – Henno Brandsma Feb 25 at 13:53

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