It does not need to be Either/Or -
there are SAKs with both corkscrew (make you very popular at parties and picnics) and Phillips screwdriver (I agree, essential).
eg: My EDC Victorinox Scientist model (customized)
Feb/1993 - present (with SackUp pouch).
Extracted from (link):
Minimalist SAKs
Phillips with Corkscrew - now that's another question.
One could get away with the
Spartan/Standard model:
shown is the 1997 100 year SAK Spartan/Standard model (the only version I have)
spartan US Model # 53151 - low price ~$16
size:91 mm | 3 1/2 inch
Large blade
Small blade
Can opener with small screwdriver
Bottle opener with large screwdriver & wire stripper
Reamer with sewing eye
Corkscrew
Toothpick
Tweezers
Key ring
Phillips? - well, the small screwdriver on the can-opener tool is supposed to fit #1 & 2 Phillips......
that's why I said: "get away with".
But a real Phillips driver with corkscrew means this model:
Scientist
from a Victorinox sheet (circa 1992/3)
This model (now very sought after) has been my EDC since 1988.
The current thinnest model with BOTH corkscrew and Phillips is the Explorer -
but it is 4 layers/backsprings thick - a bit too thick for my taste.
Welcome back is the Champion Plus model - but that's even thicker.