I read your long post a 2nd time because I am stuck waiting.

A few more thoughts:

If the “over-fire limit switch” is kicking it off
because the gas burner heated air is too hot (over 155 F typical)
the airflow generated by the
downstairs belt driven
( squirrel cage type maybe) fan
is too low.

This low airflow could be caused by.....

(A) too many air registers around the restaurant are manually closed

(B) in the squirrel cage fan the numerous blades are clogged with dirt/lint/grease.
These blades are little airplane wings and their tops and bottoms must be bare metal shapes, not “crud mounds”
This is a tedious but important cleaning chore

(C) a replacement air filter may have gotten loose,
blown into the unit guts,
and blocked off most of the AC evaporator coil fin slots
or heating tubes,
interior duct elbow,
etc

Flame sensors are frequent trouble causers of burner shutdowns.
The flame sensing designs that use the
“Ion Direct Current micro-amps”
between the ignitor and electrical ground are notorious for being unreliable.

Relays are not supposed to buzz