LMH, you are reading into my thoughts what is not there.
Samuel is not a villain. He is the prophet and the priest and the judge. But, he is not infallible. The Scripture does not imply or insinuate that Samuel was late. It says so explicitly. That did not make him culpable for Saul's actions.
Samuel gave Saul instructions on what to do when he would go to Gilgal to inquire of the Lord. Samuel instructed, committed even, to meet Saul within an appointed time. Saul went to Gilgal, and Samuel did not keep the appointment. He was not merely fashionably late. He missed his own target altogether. It was after Samuel failed to appear within the appointed time that Saul sinned, and after Saul sinned, Samuel showed up.
I am not justifying Saul. I am searching out the Word. Two things appear to me in this passage.
1. Samuel was late and had he been on time the text gives us no reason to believe that Saul would have sinned here.
2. Saul's sinfulness within existed regardless of his acts; Saul acted from what I see as cynicism about God (some may say impatience or presumptuousness, and that is true too). Saul never truly believed in his heart that God was who He had said He is. Two of the signs from God described by Samuel in chapter 10 were given to Saul immediately when Samuel told him he was king; this was the third sign -- that at Gilgal, God would meet and direct. Between the end if the early two signs to Saul and ch 13, Saul charges ahead in control, putting fighting forces in place, advancing his son Jonathan even. Time lapsed. Saul whipped some folks. Only, when the time came, and Philistines stood up in real force against him, Saul went to Gilgal and sign number three of God's faithfulness awaited. Saul took matters into his own hands when Samuel was late/delayed/had donkey trouble/lost his sundial charger and track of time/whatever. Saul was unconvinced that God truly selected him and that God was with him and for him, and he tempted God when he jumped ahead of Samuel at Gilgal.
Now, why do I say Saul was cynical? Because this account is tucked tightly in between a brief flyover of Saul the newbie king who NEEDS signs in the first place (Samuel never gave David signs) and the much longer account of Saul's systematic, absolute disbelief in God (paying lip service only when he needed favor) as contrasted to David's long patient, praising, believing heart while he was chased all over the countryside like a fugitive criminal by Saul. David who had no signs to prove God's choice of him, David who had ANTI-signs, if anything, at Saul's hand but whose belief in God endured and recorded for us so many Psalms in that circumstance.
This is all I am saying.