These things are within, and subordinate to, the legal system.
Small point: They are not within. They are private arrangements in addition to the legal system. They are creations of contract between individuals and represent the wills of only those individuals. The legal system is created by legislatures and legal societies (bars), bound by ethical rules, and are representative of the people generally.
The distinction is important. People should not think that arbitration committees or VISA's dispute resolution process is in any way part of the legal system. The legal system belongs to the country, to the democracy. Arbitration services belong only to themselves.
You can sue anyone for any reason at any time, period. The degree to which you might succeed depends on laws. To the extent a contract demands arbitration it's because it is a contract, which is defined by the legal system. The system of arbitration depends on those laws to be effective, and the backstop for it is always the courts.
This isn't true in the reverse. Congress could outlaw or limit arbitration tomorrow. No arbitrator could dissolve Congress.
Also, saying that there is no utility because in some edge cases, a traditional system is required, that's like saying a computer is useless because sometimes you need to have things on a paper.