[girigiri]

Very little to comment on in that passage ゆりさん。

servers seem to have pride of their company. - I would use staff rather than servers; "in" rather than "of" (and maybe "work" rather than "company") "Servers" is understood, but not a natural choice.

M&C Café is run by MARUZEN book chain store. They say their first president named Hayashi is the first person who made Hayashi Rice.
M&C Cafe, run by MARUZEN book store chain, claim that their first president, Hayashi, was the first (person) to make Hayashi Rice. - Only "is←was" as a mandatory correction.

Getting nit-picky:
I ate Hayashi Rice at M&C Café today.
Hayashi Rice is similar to beef stew, and an ordinary dish in Japan.
At an M&C Cafe today, I had Hayashi Rice. It is an ordinary dish in Japan, similar to beef stew.

Many restaurants claim that they have the original Hayashi Rice, and M&C café is one of them. NO ERROR.
Many restaurants, M&C Cafe among them, claim that theirs is the original Hayashi Rice.

It’s very nice place to have a break after shopping.
"a very nice place" (probably a matter of local dialect) have←take;