I actually did this too, with a lot of different remotes, in micropython on a plain pi pico -- quite fast enough.

Different remotes use different formats, a lot of them repeat the actual data twice. Most interesting were the remotes for our minisplits, which tranmit the whole state for the minisplit, followed by a second message indicating half degrees -- presumably, older minisplits could only do whole degrees so they added a second message.

Transmitting the words is more challenging, the IR LED needs to be pulsed (modulated) around around 38 khz, and there are potentially different wavelengths. So far have not been able to get our mini splits to accept a code I send to them, I suspect my LED wavelength is wrong.