There was at least one late Enigma derivative, the Russian M-125 Fialka, that did modify the reflector to allow a letter to encrypt to itself.
Curiously, rather than simply making the cipher alphabet size an odd number (which would've been natural enough for Russian, with its 33-letter Cyrillic alphabet), the designers of the Fialka instead used a rather complicated (and, for its time, advanced) "magic circuit" to replace two wire pairs with a single "plaintext enable" signal (which caused the plaintext input to be output unchanged) and a set of three wires connected to a transistorized "binary rotator" that mapped an input signal on any of the three wires to an output signal on the next wire out of the three, as in (1 → 2, 2 → 3, 3 → 1).
Of course, the magic circuit also meant that the Fialka lacked the self-reciprocal property of the Enigma: encryption and decryption were not (quite) the same operation. Thus, the Fialka also needed an "encrypt" / "decrypt" mode switch that simply swapped two of the three wires going into the magic circuit.
Quite why this feature of the Fialka was implemented in this specific manner is, as far as I'm aware of, not known. One might suspect that the original Enigma-based design of the Fialka, with an even number of letters and an Enigma-style reflector, may have been finalized before someone became aware of the cryptanalytic significance of the Enigma never encrypting a letter to itself, and that this flaw was fixed in a manner that minimized changes to the rest of the design. Or some Russian cipher machine designer might have just been eager to play with fancy new transistor technology.