I would change pedals before anything else. I just bought a cheap set of Shimano SPD pedals & cheap recreational, lace up, Shimano SPD compatible shoes. They are not as stiff as racing cycle shoes, but much stiffer than regular shoes. They are very comfortable to walk in while still being very efficient to cycle. These are my every day training shoes as well as touring shoes. Good fitting cycle shoes & clipless pedals will greatly increase your efficiency. I won't argue percentages, but I know I pull up on the pedal stroke almost as much as I push down. I know this from recently riding my Cyclocross bike with resin pedals. My feet kept coming off. I wouldn't worry too much about tire width, but I would be concerned with how well they roll (rolling resistance). I have 32mm Cyclocross tires on my San Jose & I can't tell the difference between those & my skinny tires on my fixed gear. I can ride in packs doing 30mph with no problems on the wide tires. You may find yourself less fatigued riding on wider tires with slightly lower pressure due to less road shock.
I would be most concerned with training, however. You may want to build up gradually & then try doing at least two long training days in a row to get used to recovering quickly.