Having now read through the whole of this thread, I'd like to make a few comments/suggestions if I could.
The PAPA format of having to put together a string of games in a single attempt to create an entry is used in Europe. But mainly in a Classics side tournament. Play 3 games from 12 - that's your entry, unlimited (paid for) attempts. It works fine. The better players will play numerous entries to try and qualify, the lesser players may play just their initial free ticket.
The queuing system used at many of the larger tournaments (LPC, EPC, BOp (as above) etc.) works so smoothly that I can't understand why it's not used worldwide for all major comps.
It works like this:
Every competitor has an entrance card. Every machine has their own card as well, which is kept on a large pinboard.
When a player wishes to play a machine they hand their card to the "gatekeeper" who then swaps the player card for the machine card and replaces back on the pinboard.
The player then takes the machine card to the machine and hands it to a scorekeeper who starts the game by pressing start and takes the machine card off them. (They could also take a ticket if a paid to play event)
At the end of the game the scorer logs the score and returns the machine card to the player.
The player then returns the machine card to the gatekeeper who gives the player back his card to allow him to choose another machine.
This solves multiple issues.
There is no chance whatsoever of a walkaway as the player will not get the machine card back to then swap for their card and play any further machines.
By simply looking at the pinboard you can see where you are in line and don't need to stand/sit in a queue, but can explore elsewhere. You can also gauge how many people are in front of you or which games are more popular.
As for the actual format their seems to be a number of factors which are all being lumped into one.
both PAPA and HERB are effectively pump and dump, due to unlimited paid entries allowed.
So the argument about richer players having an advantage doesn't hold true.
The difference is about consistency. A top player is more likely to be able to put together a decent string of games than a lesser player - on a consistent basis. For a single game, the top player is still more likely to post a better score, but the lesser player is more likely to post a competitive off score.
Obviously it also takes less time to play a single game, rather than a string, meaning that more single attempts can be made.
The HERB format is just not popular in Europe. I cannot think of a single tournament that runs it consistently, other than selfie leagues ran over a number of weeks where it's the only option.
The majority of score based qualifying tournaments consist of a set number of single games played on x out of y games, maybe with the possibility of a joker. EVERY game counts (unless you use a joker). It's still the top players who qualify every time, but it's a shorter format allowing more entrants on a limited number of machines in a limited time scale with less queuing.
The downside to this is that there isn't a huge amount of prize fund generated. VERY few competitions award any prize money at all. Whether that would be an issue for US players I don't know, I'm guessing it's more than likely that there are only a dozen or less players who would be winning a significant amount of prize money on a regular basis anyway and this would effect.
A compromise could well be that stick to single game entrants, eliminating the string of games which appears to be where the complaints are coming from, but rather than unlimited entries there is a limit to the number of entries that can be bought.
Say you have qualify with your best 6 games over a bank of 12. You are only allowed to buy 30 tickets in total. Do you focus on just 6 games, or spread them out over all 12.
It'll still give the advantage to the better players who can post a top score at first attempt, who wouldn't need to buy extra tickets, or they could use them for extra practice on other machines.
It would also lessen the queues.
It may also encourage players who don't usually buy multiple entries because they fear they are going to be "out spent" to buy additional entries knowing that everyone has a limit to the number of games they can play.
At the end of the day it is down to the TD for whatever tournament to decide on the format they want to use. If people want to enter they can, if they don't like the format they won't - it will be very clear in a short time which is the case.
I've discovered that (as demonstrated here) that people will complain/voice their opinions about any format, or location, or time, or the colour of the carpet, or anything else.
You can't please all of the people all of the time - although it just sometimes feels like you can't please anyone ever.
If people feel strongly enough about the PAPA style they will organise their own tournaments using the format of their choice. that's what I've done in the UK.