They started running into problems when players were getting cup tied because they played in one competition and then the selling club dropped down a league e.g. from the Champions League to the Europa League.
That meant your new signing was suddenly cup tied from the Europa League, even though he wasn't at the time of signing.
With so many teams in European competition these days, and so many teams moving between Champions/Europa/Conference, they decided just to scrap it.
They scrapped it in the English Cups as well I think. In the days of transfer windows as opposed to signing any time during a season, I'm not sure it's really needed.