Hello friend. It is true that documentation could have been better organized and have everything in one place for better browsing.
For your use case, you have two options:
1. Use the hydra API to make HTTP calls in order to register new clients. You can see the API documentation
here.
2. You can use
ory hydra client library which is compatible with javascript/typescript. Unfortunately, there are not many examples written in javascript/typescript. What you can do is have a look at the
python examples and then try to achieve the same result with the typescript client. It shouldn't be very different in terms of functions etc. The client has support for both admin and public calls, so you can use it on both sides. It can be a bit "weird" to setup but it does really work. This is what I am using at the moment.