You first look into the direction the cara are coming from (left in your case) and wait until you can cross the road. Then you check the other side just in case and while crossing the street, you look in the direction the cars are coming from again just in case something crazy happens.

If you look right first and then left, there night be cars coming from the left in which case you'd have to wait until they pass, then look right and then left while crossing the road. So your proposal would lead to one extra "looking to the side" if there happened to be a car coming.

The point of the second look left factors in the uncertainty that creeps in when you look right and spend time on the decision making process. It doesn’t feel like much but a lot can and does at times happen in that moment. You also lose some focus on what was happening on the left side because you need to shift your attention to the right. It’s like a quick refresh because once the decision is made to cross you can’t always take it back. As for you assessment of it being redundant I get where you are coming from but it’s more of a percentage of effectiveness thing. The first look left might only add another 5-10% to the the accuracy but that’s where most the errors can be prevented

You're presuming people are paying attention to everything all of the time, which they don't. Humans need that extra level of redundancy to mitigate our own shortcomings. Also, we look twice right as well! Once while standing still, the second time instinctively when you're already crossing in case something changed. Would you be required to do this 100% of the time? Probably not, but learning from a young age you can't trust one look is good practice in more ways than one.