I got my Toro MX Timecutter back in 2014 New. Have the Kawasaki engine, & no heat shield under my muffler either.
It has never burnt the grass under it. But when I start mowing I don't stop, but do park in same place to blow it off after & wipe it down..
Still no burnt grass. I'm thinking with newest emission regulations, yours is running extra, extra lean, thus making it run hotter.
I would bet your spark plugs are snow white on the porcelain.
What engine ? Does it run good ? Cause if it has an intake leak, it will cause a lean condition, & should be corrected asap.