If you maintain an app that offers the same core functionality as the website, and the user can accomplish their Immediate goal on the website, adding the unnecessary step of transferring over to the app is never a superior experience. Simple as that. The only valid case for prompting a user to open your app is when their goal can't be accomplished on the web. Anything else is promotion serving your goals, not the user's.

Don't accuse parent of projecting for taking your words at face value. You stated development budget as a reason for wanting users to download your app. That has no relevance to any user's needs.