It doesn't always take a long time to improve standards. Ten Hag and Emery have done it within weeks.
There are loads of examples of managers who have improved more than one team significantly within a few months. Some of those guys were available and at least one said he would have come (Favre).
If you're right, and we must expect a long wait before seeing any improvement from a new manager, then it's a terrible idea to wait until Feb/March to see if things get better once the imaginary striker arrives. A new guy would justifiably ask what we expected in a few weeks.
The argument that we shouldn't expect better because he's a relative rookie doesn't sound like a strong reason to keep him to me.