As an update on some of my experimentation, I've killed him only using a total of 5 crossbow bolts...
As sharpshooter or support spec? Solo or with others?
A 1 player suicidal patriarch only have 7000 hp, so if you were using sharpshooter it isnt really that strange. In theory you can take him down with two arrows, one to make him kneel (4050 damage) and a second arrow while he kneel, before he stealth and run away to heal (total 8100 damage > 7000 hp).
Crossbow deal 1800 headshot damage for non-sharpshooters. Patriarch kneel when he lose 25% of his total HP. He need to have more than 7200hp (25% of 7200 is 1800) or he will kneel form your first headshot. A one player suicidal Patriarch have 7000HP (4000 base HP times 1,75 for suicidal = 7000).
If you can stop him from healing then, in theory, you only need 4 arrows to the head as a non-sharpshooter to kill him in a solo suicidal game (1800*4=7200 which is more than 7000HP).
This has nothing to do with having an "extra weakness" in his "eye" or not mind you. To verify if he got some strange "weakness" if you hurt his "eye" i went into a beginner game (where he deal less damage so i can go close) and shot him with half a clip of 9mm rounds to the "eye" at short range before he used his rocket and another half clip to his "eye" during his chaingun animation. He did not kneel.
As a non-sharpshooter you will kill a solo suicidal siren and bloat with *one* arrow to the head (she got 300 base HP times 1.75 = 525HP and he got 525 base HP times 1,75 = 919 HP both are is less than 1800 damage).
As a non-sharpshooter you will even kill a solo suicidal scrake with *one* arrow to the head (he got 1000 base HP times 1.75 = 1750HP which is still less than 1800 damage).
As a non-sharpshooter you will kill a solo suicidal fleshpound with *three* arrows to the head (he got 1500 base HP times 1,75 for suicidal = 2625HP, however non-explosives deal half damage so you only deal 900 damage. 900 times 3 = 2700 damage. One arrow to enrage him, one arrow when he enrage, but still stand still before he charge and then a third arrow when he get close enough so you cant really miss).
Crossbow headshots (if you know how to aim) are just that strong. Shrug.
The problem with aiming for headshots at long range on higher difficult settings is that you pretty much need someone to bodyguard you. Taking out them gorefasts, crawlers, stalkers, clots and bloats while you aim. Bullpup Commandos and Xbow Sharpshooters make a really strong combo.