I agree that there is an added psychological element involved in a large one-group playoff. However, although I think the added psychology very well would make the playoff more interesting, I don't think it is a necessary addition. In my view, any advantage gained or lost by being in either twosome is not large enough to determine the outcome (did any of the four players have significantly easier or tougher conditions to endure in that playoff?). It's not like the R&A said: "Thomas and Steve, you will tee off now, and Stuart and Ernie, you will return tomorrow to tee off at 7:00."
PGA Tour events have had playoffs with four or more players (witness Riviera last year) and they all played together because splitting them up does create an advantage whether determined by "lot" or any other means.
Matt, are you sure that is the reason that the PGA Tour puts everyone in the same group in playoffs? My gut says their reasons are probably more related to spectators/TV ratings than ensuring fairness. If they were all about fairness, they'd have random pairings for Thursday-Friday instead of the A/B/C player system they have now, with rookies and qualifiers relegated to 1st/last tee times of the day slots.
I should add that there is a big difference in a sudden-death playoff than a four-hole aggregate. Remember, in a four-hole aggregate every player who starts the playoff finishes it. In sudden death, especially with a large group, the numbers of players is likely to shrink with each hole. It wouldn't make sense to split up the groups in a PGA Tour sudden death playoff, with the first group having to wait each hole for the other group to finish, then perhaps regrouping for the next hole.