Grazing rye as a grass path?

Not sure about the feasibility of what you suggest, but if it is an annual that generally means it germinates, grows, flowers and sets seed all in the one year.  The specific plant does not survive more than one year, but any seed which lies on the ground may start a new cycle the following year.  If you plan to mow before it sets seed, it will not live into a second season.

Having said that, I'm not familiar with the plant grazing rye so someone else will hopefully be along to give you some more pointers.

Seems to me that if you mow rye, it will be stubbly like a farm field and not pleasant to walk on, least of all with bare feet or sandals.