Snell had rarely or never raced a 1500 before the 1964 Olympics. In his book he remarked on how nice it felt in Tokyo to just get settled down from the start of a 1500 and find there were only three laps to go.
As pointed out above the Commonwealth Games during Snell's time were raced at the imperial distances. (Tokyo was the only major championship in which he raced 1500.) Snell was possibly in shape to give the 1500 record a go on several occasions when he ended up racing a mile instead.
And that "racing" was key for him. In contrast to Elliott's sustained power Snell relied on his sprint finish to win championships. Snell's 1500 in Tokyo also followed his rounds (and win) in the 800 and that meant that he'd not be well served by a potential-WR pace in the 1500 final.
https://www.youtube.com/watch?v=RK0woLEtQQA&t=12s
Watch Elliott's Rome race again. He crushed the field from 800m to 1400m and then was buyin' it big-time on the homestraight. If anyone had stayed close enough to him (but no one could do that in 1960) to muster a sprint then Elliott would have been vulnerable.
https://www.youtube.com/watch?v=Wd2ISNCxFoM
But yeah--Herb's 3:36.0 and then his Rome 3:35.6 were just too far ahead of all competition available at that time. Snell had had a bad experience in his second WR mile (a strict pace run and not a real race) and probably wanted no part of another specific record attempt. And for Peter to beat Herb's 1500 time a targeted WR attempt would have been necessary.