Not exactly. The banks do not want less deposits (ask SVB how bad that can be!) but they try to get away with paying as little as possible and keep the spread. (You wouldn't say that the main objective of companies when they raise prices is to have less customers, would you?)
If they wanted more deposits, why would they pay as little as possible to 'keep the spread'? No bank is anywhere near monopoly; there are tons more deposits to get (if you want them, and can make money safely on a spread).
For the same reason that many businesses do not want "more customers" above anything else - to the point of selling at cost or at a loss - and if they increase prices is not necessarily because they want "less customers".
Some banks actually do want less deposits. They do things like increasing fees to chase them away. But it takes some real discipline and investor management to pull it off.
