Websites are better than apps in this context.
They have broader compatibility, and are usually easier to use and quicker to download.
As a user, why make me download an app I may never use again when there is probably a competitor that doesn’t put such a burden on me? Unless you know you have a dedicated audience of repeat customers, most patrons will only have one visit. Push notifications and other app features are potentially overkill when all they want is food.
Menus (even online digital ones) are familiar with an expected interaction model. A brand new app won’t be familiar and it will take a lot of work on your part to make sure users don’t spend several minutes fiddling with things and slowing your turn times.
You mention that this is a progressive web app, but also mention using an app store. Using an app store comes with additional complications.
Are you 100% sure all your patrons will have compatible devices? You might be reasonably sure they have smartphones (and can access websites), but what is the app store’s lowest supported version? Suppose you seat a large family and all they have is an iPhone 4. What do they do? Fall back to paper menus and cash only? What if they don’t have cash?
Making payments in an app also usually requires you to give the app store owner a cut, often 30%. Restaurants don’t usually have high enough margins to cover that.
A possible way to make an app more viable would be to incorporate it into the restaurant experience to make it worth downloading. Add a live video feed from the oven so I can watch my pizza bake, or let me message my waiter and ask for more bread without needing to catch them walking by. It might be a novelty, but whole restaurant chains have been built around such things.