Would setting the time to the Mushroom Session (Late Fall) and finding a Elegant/ famous mushroom be quicker then doing the normal rout?

Not likely to be honest. You're more likely to find Perfect fruit than fungus (even though a Famous [16k], Elegant [10k], or Rare [16k] mushroom is potentially more diverse possibility-wise), which doesn't spawn nearly as consistently as fruit. Very intriguing idea though. If you give this a shot and it works out, it could change things quite a bit.

Now that I'm typing my thoughts out, I feel like hunting around for specific mushrooms in Mushroom season would potentially waste more time than save via fruit trees you'd encounter in your normal pathways as it is.