I don't have time to give a course on modulation
Don't even try please.
"the command data is a Manchester coded bitstream modulating a 36 kHz carrier". As you see the carrier has nothing to do with the optical frequency.
When you properly tell what actually is carrier - then sure. Look... I do not deny possibility of signal double-modulation, nor possibility to transmit QPSK-modulated baseband using amplitude modulation of IR light. What you propose is indeed possible to build, but it is pointless overengineering. It is already said multiple times here -
pointless.
NB. In the standards I'm citing only amplitude modulation is used but there is no reason a phase modulation (BPSK, QPSK etc.) could not be used as well.
IR remotes do not use amplitude modulation but OOK (on/off keying), even for "modulation of the carrier". BTW IR receivers have pathetic selectivity - if you point interfering (transmitting) 56KHz remote to 36KHz receiver, it can't decode 36KHz transmission aimed at it. That "carrier modulation" for
TV IR remotes does not create parallel transmisison channels, it is just smart balance between design complexity and
protection against false triggering from ambient IR noise and other kind of
TV IR remotes.