Reused menus were always a gross idea so I'm on board with this.
That said, menus spend a lot of time on the table, some of that while food and drink is being consumed - apps/drinks are ordered and people continue to ponder their meal choices as they munch. The drinks splatter, the food drips sauce and grease, etc.
Because of the visibility factor, you're confronted with this information far more than you would be for a doorknob or chair. That's going to cause some to consider the menu more gross than those other features.
Finally, tables and chairs and doorknobs and railings all get cleaned from time to time, with tables and chairs often wiped down between guests. Paper menus don't take well to being wiped down with water or cleaning solutions. You can laminate, but laminated menus tend to be associated with cheaper / lower class establishments.
At a place with a continuously changing menu this is less of a factor as menus are reprinted often, sometimes daily. When the menu is static, paper menus might stick around for a while, accumulating residue and getting more visibly dirty over time.
Everything could be gross to touch if you really think about it too much. Your phone is probably worse than any menu.
Not just any website. Often it sends you straight to some online ordering portal like Toast or Doordash. And this bring all the slow performance and pop-ups and adware that implies. Very fun to try to load these bloated web apps when you have 1 bar of cell signal.
I imagine most of the business in such places comes from delivery so they don't care much about dine-in guests.
When it's a website, it's often lousy long lists where you can only see 3 or 4 things at a time, have to scroll super far back up and down to compare things, and have no idea what the restaurant is nudging you to order.
Everyone forgot about digital menus fairly quickly after it became clear that covid doesn't spread through surfaces though.