The problem with Bluetooth is it has very unstable bandwidth and it dips down to the range of 100kbps. So what you need is a codec that can adapt to bandwidth changes very fast. In some situations, with latest Bluetooth, you could even stream FLAC directly, but that requires nearly perfect conditions, and it would start stuttering as soon as that changes. So for that reason, AptX HD is designed in a way that it switches between regular AptX and lossless depending on the connection strength.
Most of these are proprietary, so there isn’t much to base on. If you were going to write an open source alternative codec, you would basically have to do it from scratch.
And yeah, these SBCs (nor any other that I know of) don’t have any Bluetooth audio-specific hardware. They just have a Bluetooth transceiver and what you do with it is entirely up to the installed software. By comparison, DAPs usually have a dedicated hardware chip designed to decode Bluetooth audio. That’s what decides the supported codecs there.