This is solved now.
I tried 1) emptying the browser cache & cookies, 2) putting Brave’s settings on “Privacy and security” to “Allow all cookies”, and 3) “Shields” settings from “Block cookies” on disabled, but it didn’t change anything.
Then I noticed that “Trackers & ads blocking” was on Standard, and that the other options were “Aggressive” and “Disabled”.
Once I changed the Settings – Shields – Trackers & ads blocking from “Standard” to “Disabled”, the cookie consent pop-up started to display normally. And they began to show up on other websites as well.
I guess I’d gotten so used to not seeing cookie consent pop-ups, that I didn’t realize that Brave doesn’t show them even on the “Standard” settings (one could assume that the “Aggressive” mode was for that).
So this is not a plugin issue, but a Brave browser’s feature. Hope this helps someone who might be puzzled by the same situation.
Thanks for the awesome plugin!