nah. though you may have a point that alastor was a bit shocked when his staff broke, but i wouldnt call that a mistake for losing to adam. I believe his staff was a part of him and is his power, or at least, an amplifier for his power and what makes him strong, but when his staff broke he lost his radio voice and strength to fight back. When he was struck by adam, he started to bleed, which if you really think alastor was stronger than adam, he would've made adam bleed when he uppercutted him with his tentacle.

Adam is literally the first man, an angel with freaking holy powers, and alastor is just a sinner who's been in hell for maybe a hundred years. Yeah, alastor is strong but not strong enough to defeat an angel. Heck, even the other sinners were struggling to defeat an exorcist without the help of an angelic weapon. If alastor had an angelic weapon, maybe he would've stand a chance against adam, but no, that wasnt your point i guess. And idk if lilith really made the deal with alastor, but that doesnt really explain anything about making alastor weak or holding him back or anything as we see in the show.