In the first few years of your education, yes they are open to all. However, once a child has gone through several years of french immersion, they have the ability to choose between continuing in french, or dropping into english. A child who has done only english has a similar option, but it will require considerable extra work on their part to switch to french. So a public french immersion high school cannot be considered open to all.

>care enough about their kids' education (specifically knowledge of French)

Not specifically. Certain public schools can be more well funded, or specialize in different areas than others. A school which has a french immersion program might also have a heavier focus on academics than other schools. Once it gains a better reputation, it might become attractive even to those who aren't interested in having a better education. A french immersion program is a way of limiting such peoples access to these schools.