Well, people used bike shoes back when clips and straps were the only deal around. Riding in sneakers will never be as efficient as bike shoes.

Having said that your clip placement is wrong and could be too short. The clip should not be over the big toe but more toward the center of your foot. Is the ball of your foot over the pedal spindle? If you can't get your foot that far into the clip then it's too short.

Bikes that come with clips and straps tend to use cheap clips and straps since these pedals are frequently swapped out. Used to be that clips in different sizes were readily available but I don't know if that's true anymore since relatively few people use high end clips and straps anymore.