Come Fly With Me solution
by Kegler

As indictated by the center entry, BYE BYE BIRDIE, eleven of the answers contain birds (shown below in red) which are removed before entry.

Come Fly With Me solution grid

Across
1DISROBING (RIGID SNOB anag)
4ACKNOWLEDGE (WE GLANCED, OK anag)
9MI + C(ROW)AVES
10H(OLD)OVER
12PI(G-MEN) + TED
13SAT (1st letters)
15EDWINA (A NEW ID anag)
16E(WE)R
17T + ILLS
19R(AD)IO
20RECOIL (2 defs)
21BYE BYE BIRDIE
24S(HIES)T
27FEINT ("FAINT" hom)
30DE(CO)Y
32ALTO (hidden)
33DAM + I + AN
35LYE (hidden)
36CREDULOUS (USER COULD anag)
38YON + I
39TEN + ETS
40GUT + W(R)ENCH + ING (TUG rev; GIN anag)
41MALAR + KEY
Down
1DAPPER (2 defs)
2SIGN (2 defs)
3GAM (MAG rev)
4AVE + RT
5CENTIME (TEN MICE anag)
6NEED + LE
7DO-SI-DO (ODD SO I anag)
8enGLAND
9MID + WAY (YAW + DIM rev)
11REGRETABLE (REBEL TARGET anag)
14B(RIB)ED
18S + CREAM
21BE(MUSE)D + L,Y
22B(ROAD)EN
23INTRAVENOUS (RUNS TO A VAIN anag, &lit)
25INTER + N(MEN)T
26SE + ARCH
28T(OAST),Y
29BAYOU (A BUOY anag)
31YOUNG (GUY ON anag)
34BO + TS + WANted + A
37LEM (MEL rev)