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The diodes clip the signal, not the op-amp. Op-amps only clip when the signal reach the power rails (or close, depends on op-amp), and it sounds very bad, noisy, squeally and probably oscillate like crazy (been there before). The diodes limit the voltage and thus clip it. That's how the TS generates its overdrive sound. You can use pretty much any dual op-amp for the TS without any noticable change in sound.
Any dual op-amp will do the job for you. If you already have a JRC4558, use it. If not, TL072 are very good and quiet op-amps, and they are very cheap. I use them all the time for building overdrive pedals.