If you are simply renting an airplane to build time, not undergoing instruction, it is a bit of a strange policy. Presumably you have demonstrated your pilotage skills when you took your checkride and/or checkout. So now that you are simply renting, how you navigate, be it ipad, pilotage, kentucky windage, whatever, is basically your business IMHO provided it is done competently. Essentially I would be expecting to be treated like a grownup, otherwise I would look for somewhere a little less rigid to build time.
That said, plotting courses on paper and such is good practice and will make you a better pilot. Plus it's more fun to look out the window and identify landmarks than to tap tap tap on your ipad. We have enough gee-whiz distractions in our life, enjoy the simple flying while you can.