You can request to change your primary care doctor. There is an online list of Kaiser doctors who have capacity and you can choose to change to them... the list is accurate and you can do it online, though, unlike those described in the article. You can also make appointments with other Kaiser primary care doctors, either because you want to see someone else or your doctor doesn't have availability.
You don't always have a choice to see a different specialist if you live in a smaller town, but in larger cities you can request a different one.
In general you can't make direct appointments with specialists on your own, you need to be referred by your primary care doc. Once you have seen a specialist, though, you can make direct appointments with them.
You can also call and tell them why you think you need to bypass the normal system. They have staff who are there to help navigate those situations. I have never had to do it personally, so I can't vouch for how accommodating they are.
I will say, though, that in my experience I have a lot more say over my care than my wife does with her traditional insurance provider. She has had to fight to get things paid for and been rejected way too many times.
Sure, she could make an appointment with any specialist she wants... but will insurance pay for it? In your example, if you have another doctor who has said the problem is just anxiety, will insurance pay for your second opinion?
If insurance isn't going to pay for it anyway, it doesn't really matter if you have Kaiser or a traditional insurance provider. Both of us can make any appointment we want if you are going to pay it yourself.