There are currently 26 NHL goalies who have faced at least 100 shootout attempts.
Confirmation bias dominates one's reading of the list; the person who follows the natural assumption that some goalies are better at shootouts than others will point to Pekka Rinne being first on the list and Henrik Lundqvist being fourth as supporting evidence for their argument. The person who is out to prove that shootouts are random will point to Johan Hedberg being second on the list, Mathieu Garon fifth, and Rick DiPietro sixth.
Ironically, both sides probably think that Marc-Andre Fleury being third supports their point of view.
So instead of doing this anecdotally, let's look at the data.
Here's the relationship between career save percentage and shootout save percentage:
Save percentage during play is worthless as a predictor of the shootout; Rinne and Lundqvist are the only goalies to have a really high save percentage both in normal play and in the shootout. So pretty much every team but two has had to come to grips with the idea that their goalie won't be great in both situations, and I can promise you that the fans of the Devils, Lightning, and Islanders don't consider "good in shootouts" much of a consolation prize.
Moreover, Ilya Bryzgalov isn't even particularly bad at shootouts. At 63.2%, he's slightly below the 67.7% league average (if you were wondering, that dot down at 56.3% is Niklas Backstrom). Yeah, giving up 8 goals on 10 attempts this year is atrocious, but the fact that he could be a 66% goalie for his career up to now and then 20% so far this year just underscores what a small sample size we're worrying about.
It is natural for a Flyer fan to focus more on the four games that they saw than the multiyear career in Phoenix, but a rational look at whether he is genuinely bad at shootouts would have to weight the multiyear career more than those last ten shots. Being incredibly frustrated that the team keeps losing shootouts is reasonable; panicking about it and calling for the Flyers to bring in a relief goalie for the shootout is not.
Bryzgalov is going to eat up a lot of cap space for a long time and hasn't been very good so far in any phase of the game. There are plenty of much more humongous big fish to fry than those four shootouts.