Unfortunately most people's primary computing devices are smart phones and smart phone radios cannot keep a TCP connection open, or won't, because of power usage.
While this might be true for users connection to services there are also a lot of intra-serices connections that can keep TCP connections open and greatly benefit from doing so. For example, a microservice architecture where you have APIs communicating through HTTP with each other.
Keeping connection open does not require any action. Also radios have nothing to do with connections. Connections are abstractions from a different layer. Radios are shut down much more frequently than you'd think, to save power.
I thought the primary use case for this is you could load all of your websites resources on a single connection so you can make 100 file requests at the start and it only makes one connection.
Are you asking why the radio has to be on or implying there are other reason that the radio could be on for? If you have a TCP connection open but the radio is off how are you going to receive incoming packets?
If it's a keep alive connection, you can just get the packets when you next turn on the radio. It's not a big deal to get the close right away; not that there's a good way to tell the system that. Anyway, you probably have a tcp connection open for the system push channel; if that requires the radio to stay on, it will be on.