I actually disagree with this view, because Liverpool won the Premier League title but were eliminated early in the Champions League. It's not true to say that a team that succeeds in the Premier League will definitely succeed in the Champions League. Similarly, remember that Chelsea didn't finish well in the Premier League the year they won the Champions League. Therefore, while the Premier League is very important, it wouldn't be right to judge it directly based on their performance in the Champions League. The two competitions are completely different. In fact, I think the Premier League is more difficult, but the atmosphere in the Champions League is much different, and the teams play there differently.
The Champions League certainly presents a different level of tension than European domestic leagues. Teams that dominate and succeed in their domestic leagues don't necessarily achieve success in the Champions League. I think this applies to all teams representing their domestic leagues. It's not just the Premier League that's highly competitive. Even PSG, which has dominated Ligue 1 for several seasons, has struggled to win the Champions League.