It does sound like entrainment you are experiencing. The hearing aids were using an obsolete method of trying not to whistle (feedback) and in doing so they sometimes confuse themselves when they hear certain noises, and ‘panic’ adjusting themselves in such a way they end up making stupid noises as you describe.
The good news is that modern hearing aids shouldn’t suffer from this problem. Those Audibel aids sound like they need some adjustment to get the best sound.
As for sound processing and directional microphones, you are not immune to those benefits, just as long as the features on the hearing aids are set correctly.