This isn't all or nothing here everyone. Two things can both be true. 1) 714 or 626 is a ridiculous achievement regardless of pocket size, and 2) Soft equipment makes a huge difference. BOTH ARE TRUE.
OP is right. Big pockets have three big advantages:
1) Pocketing. Many runs end with a miss, where a ball wobbles and doesn't quite fall. This simply happens much less often on soft equipment. Combination shots, balls down the rail that wobble a bit, just stuff that's a little rough. When these things just suck in with a shallow shelf it's a breath of fresh air. I've heard it said that most players that ran 100 actually ran 30 a few times connected by balls that fell in that shouldn't have gone.
2) Cheating the pocket. YES YES YES. OP is right on. With big pockets you can cheat freely, and this really increases your control over the cue ball and your ability to create angles needed to nudge around other balls. If you want to get an idea of how important this is, play on a regular table and play with the rule where you get to move the cue ball 2" after every shot. It's a small thing, but on every shot it adds up to a huge difference.
3) Dead combos. On almost every straight pool rack there are secondary break shots. This means you have to clear the blocked pockets, get on the break ball, open the rack again, and get the cue ball clear for a shot. When there is a dead combo in the stack you can bypass all of that and just whack the combo in at speed and rip everything apart. Well, on a tight table there might only be a dead combo in the stack 1 in 5-10 racks. On soft equipment it's more like 1 in 3 because you can play shots that you trust will go. That's a big advantage.
4) Stamina. We all know that focus is a huge thing in straight pool. In general the easier the game is, the less effort we expend, and the longer we can keep going. Even if someone was technically sound enough to run straight pool on a Chinese 8 ball table, the effort required would wear them out much, much faster. If someone is going to post a big run it has to be on a table where they can auto-pilot, they don't have to dig deep to put down testy shots, they can just throw them at the pocket and trust they'll find their way in.
Newer cloth also makes a huge difference, as does the template rack.
OK, none of this takes away from Shaw. I am totally awestruck and inspired by his performance. I've watched a lot of the run very closely and learned a bit by him. Some technical knowledge, other bits just a swift kick in the rear that tells me I have to elevate my own game and showing me what's possible.
But on my table, with fair but stern pockets, broken in cloth, triangle rack, normal pool hall balls, etc, it just doesn't work the same. I have to be realistic. I don't know exactly, but I'd expect that a run of 100 on my table would equate to a run of 150 on the softer table.
I just wish we could agree on both points without thinking acknowledging the impact of soft equipment equates to hating on John or Jayson. Now for those people who do minimize these accomplishments, that's different! Open fire on those guys!