I've been running clipless fixed and really like it. Much easier to stomp in than it is to try to get clipped pedals flipped over and tightened on the fly.

If safety is your main concern, I'm not sure how to look at it. It is pretty easy to get out of clipless pedals in a hurry once you have trained yourself to instinctively rotate your heel outwards when you need to put a foot down. Toe clips and straps are a pain for me because once I get them tight enough to let me use them for pedaling effectively, I can't yank my foot out of them quickly.

My vote would be to keep the eggbeaters on there. I'm not sure why people like clips and straps so much. Maybe it's for the image ;-)