Mildly Infuriating

The goal of childhood is to prepare you for adulthood. It is better to be teased as a developing child, especially for something trivial, and be a well rounded adult. Children have to figure out how to navigate difficult social situations themselves, rather than simply avoiding them. It is becoming increasingly problematic when kids aren't working things out amongst themselves, or at the very least putting up with it, and instead resort to having an adult fix it for them. If they learn they can always go to an adult to fix their problems, they are being prepared for an authoritarian government. The solution to their problems is a higher power that will fix things for them. This is not quite the same, but it is avoiding difficult confrontation over something as trivial as shoes.

I do think kids should also have the freedom to choose their own shoes. If you give them a budget, and they can find Nike shoes in that budget, good on them. Maybe they even keep an eye on them going on sale. But if they cannot find shoes within the budget, they will have to settle for what they like within their price range. Which is also a valuable lesson for a developing teen.

they feel those problems with the same intensity we do

We cannot protect kids from big feelings. It is vital they experience big feelings. It's becoming increasingly problematic with over protectionism and treating children as fragile beings. It's caused higher levels of anxiety and reduced social skills. While you may say them not having name brand shoes will lead to anxiety, if they are always given a way out of their easy to handle middle school problems, how are they going to be prepared for adulthood problems, or the countless other things out of their control. They need to experience the anxiety and learn how to handle it in healthy ways.

something that’s important to a child should also be important to their parents, in my opinion

I get where you're coming from, but that cannot be universally true (and I think you would agree). A child wanting every toy they ever see, no matter how important to them, obviously is not going to be important to you as a parent. If a teen thinks it's important everyone they meet loves them, you cannot encourage extreme people pleasing. No kid "needs" name brand shoes. That is very distinctly a want. Perhaps they do some extra chores to earn their more expensive shoes, so you are all happy. But simply giving them expensive shoes they will inevitably grow out of because of a few comments from some school bullies is not a big problem. It is a little problem. Kids can handle little problems without adult intervention.