I might think that this also applies to its roasting and rinsing; it is, therefore, written (Ibid. 2) "And you shall slaughter a Pesach": Slaughtering was in the general category (of "making"). Why did it leave (that category for special mention)? To serve as the basis for a comparison, viz.: Just as slaughtering is expressly for the (altar) service, so all such "makings" (must be specifically to the end of "Pesach"), to exclude those (like roasting and rinsing) which are not altar-oriented.
"for in the month of Aviv": a propitious month, one which is neither too hot nor too cold. And thus is it written (Psalms 68:7) "G-d settles (the scattered) individuals (of Israel) in a (single) house (i.e., a united congregation). He releases the prisoners (from Egypt) in an equable (month" [neither too hot nor too cold]).