> will raise deposit rates to attract deposits if they constantly find that they need to go to the interbank market to balance their books because it is cheaper

This is true for the largest banks. For many smaller banks, interbank lending is cheaper than deposits. Particularly if those deposits must come from new customers.

Your model is roughly correct over long, strategic time periods. But in tactical timeframes, deposits are assumed fixed or lightly. (In fact, the post-97 role of money centre banks has been to attract deposits to then lend to smaller banks.)