no advantage or disadvantage to timing, from a recruiting numbers standpoint. obviously there are psychological or game theory advantages and disadvantages to early/late recruiting.
when you ask... if the two teams all-in'd at the same time, same prefs, would the slightly higher prestige team win all the time? i am slightly confused because the next line you say, your assumption is, the greater disparity, the greater the odds. overall i read this two different ways...
what i'll say is, there is zero 'luck' outside of the 'luck' of who you happen to be battling and the true luck of the dice roll. with all else equal, the higher prestige team would be ahead in 100% of the scenarios. but if they are slightly higher in prestige, this might only shake out to a 55 over 45% advantage, or 60/40, or so forth.
the way recruiting works, and has always worked, is that when you do an action, the action is converted to a number of points, which is multiplied by the relevant multiplier, and this number gets added to your total score for that player. at the end of recruiting for this player, those total scores are compared to figure out the end situation. today, if those scores are close, you get a % chance. but there is no randomness to like, you did a HV and maybe you got 10 points, maybe you got 50. its all very deterministic.
just to note, there used to be a slight, per cycle multiplier on top of your effort, so that recruiting late was less effective, but this is gone.