There is no specific reason that these APIs don't exist. There are no concerns around security or performance that are specific to the HTTP protocol itself. HTTP can be easily secured with HTTPS (much more testing than your brokers custom solution). Traditionally HTTP has had less support for streaming data over persistent connections as the main way to do this was a chunked encoded stream of unlimited length. The introduction of web sockets removes any impediment to using an HTTP + WebSockets based solution.
A custom protocol would still need to implement error handling, checking for dropped packets, connection disruption, etc... You could implement this just fine in HTTP. Developers of custom protocols often describe efficiencies of using custom binary protocols but there are rarely accurate and detailed measurements to back up their assertions. It would be very simple to build a very high performance solution using nginx and WebSockets that supported any piece of functionality that brokers usually provide.
I would suspect that there is not more of this since most financial developers don't typically turn to these types of solutions just because that is not the way things have been done. Inertia in any industry can be very powerful.
In either case various FIX engines are already heavily tested and in use at scale and since most people trading at scale are using FIX there is little incentive to invest in developing a web based technology. If a brokerage already has an existing custom API the ROI of moving to a HTTP based solution is probably almost zero.