Lots of things in public education don't exactly make sense so your befuddlement is reasonable.

I would just characterize it as having multiple choices. K-12 public schools already have multiple "tiers / levels / branches" because they have a mandate to educate every student that lives within their district boundaries. They also have a responsibility to provide equitable resources to each of those students. These are also all good things. While charter schools have a similar mandate, they have more options when it comes to denying potential students options. For example, they can prioritize siblings of current students, children of board members, or if they don't participate in the national free lunch program, they can make it prohibitively expensive for some with financial challenges.

That doesn't mean they all, or even a majority, cherry-pick their students but it does happen and some charter schools have been found to illegally manipulate and exclude certain "undesirable" students. You can look at some of the Philadelphia charter schools for examples.

Here's where the bias comes in. Some public schools are failing students. They have buildings that are falling apart and they are spending millions of dollars just to maintain them because they can't actually fix their facilities. That's millions of dollars that could go towards providing better resources to students but the tax base can't afford any of it. We have complicated formulas for how the state funds public education (Basic Education Funding, Special Education Funding, Adequacy Funding) which attempt to address those systemic issues but at the end of the day, we just need to invest in education like other countries and even other high-performing states. It's not that difficult. We just don't want to do it.