QUIC is roughly TCP-equivalent not HTTP-equivalent, and we do have a BSD sockets API for TCP. You might be thinking of HTTP/3 rather than QUIC; HTTP/3 actually is HTTP-equivalent.
You can turn the OP's question around. Every modern OS kernel provides an efficient, shared TCP stack. It isn't normal to implement TCP separately in each application or as a userspace library, although this is done occasionally. Yet we currently expect QUIC to be implemented separately in each application, and the mechanisms which are in the OS kernel for TCP are implemented in the applications for QUIC.
So why don't we implement TCP separately in each application, the way it's done with QUIC?
Although there were some advantages to this while the protocol was experimental and being stabilised, and for compatibility when running new applications on older OSes, arguably QUIC should be moved into the OS kernel to sit alongide TCP now that it's stable. The benefit of having Chrome, Firefox et al stabilise HTTP/3 and QUIC were good, but that potentially changes when the protocol is stable but there are thousands of applications, each with their own QUIC implementation doing congestion control differently, scheduling etc, and no cooperation with each other the way the OS kernel does with TCP streams from concurrent applications. Currently we are trending towards a mix of good and poor QUIC implementations on the network (in terms of things like congestion control and packet flow timing), rather than a few good ones as happens with TCP because modern kernels all have good quality implementations of TCP.