"Let's eat in this restaurant." is now accepted.

As someone whose second language is Chinese, I think 식 is similar to the Chinese word '食' (pronounced as 'shi', which means food) because to me, 식당 sounds like 食堂 (pronounced as 'shi tang', which literally translates to food hall). So I think the word 식 is related to food

I noticed both words you mentioned are related to 음식 which means food.

Looks like 식 comes from 食, which means food and is pronounced very similarly.

Yes. It comes from 食.

https://en.wiktionary.org/wiki/食

Food related things. Just like the 학 in 학생 and 학교 means school related things. It also probably means other things on its own.

Looks like 식 comes from 食, which means food and has a similar sound.

Also, in Cantonese, eat is pronounced 'sik'