I haven’t played Flashpoint or Clash, but I can try to explain why I don’t like Push.
First, matches tended to be drawn out. While there was back and forth, since there was this time of having to run the robot back after your team took control of it, it felt like it was just padding time to the match without adding much to it. A lot of games felt like they were decided by the first team fight (more on that in the second point,) but still had to go to Overtime, meaning that it felt like you were stuck in a battle you know you’d lost, but were forced to slowly finish.
Second, matches felt snowbally. If you took the lead early it was easier to keep the lead since the enemy team had to undo all your progress before they could make any of their own, which meant if Team A won the first team fight and then Team B won the second one, Team A still had the advantage. It is, as far as I know, still the only game mode that gives a distinct advantage to the first team to win the team fight that no other game mode does, since if Team A wins the first team fight, it actually increases the amount of time Team B has to control the point. There was a video about it, that pointed out things like if one team manages to control the bot for 106 seconds, the other team has to control the bot for 128 seconds to take the lead, meaning they’re penalized 22 seconds because they didn’t hold it first.
Imagine a KotH map where one team has control of the point for 110 seconds and the other team has control of the point for 120 seconds. It’s clear that the 120 second team should win, right? Well consider a push map where one Team had control of the bot for 110 seconds and the other team had control of the bot for 120 seconds. Which team won? The answer, unfortunately, is it depends, since if the team that controlled the bot for 110 seconds controlled it first, then they win. While if they got it later and the team that had it for 120 seconds won the first engagements, then the 110 second team would lose.
Third, the mode had perverse incentives. At some points, it actually became a better idea to not further the objective, because that would give the opposing team a better chance to get into Overtime. If you had a lead on the enemy team but the bot was way behind you, your team could just intercept the enemy team trying to run to the bot and pick them off. If you could stop the enemy team completely until time ran out, you’d win right there. If someone slipped by to go touch point, since they still had to push it for a while you could 5v4 the enemy team and win, then just go pick off the one person on the bot. But if you were trying to play the objective? Then the whole enemy team can hit the bot at the same time, getting the chance to win an even team fight without having to worry about rushing to it to avoid overtime.
Finally, I didn’t like that the robot moved the same speed regardless of how many people were on it. With the payload, it moves with 1 person, moves faster with 2 people, and moves fastest with 3+ people. That gave your team an incentive to stick together on the payload to keep it moving faster. But with a constant speed no matter how many players are on it, a lot of teams just left one person to babysit the bot while the other 4 ran off, which was just boring for the babysitter. So either you ignored the objective after you took a big enough lead, or someone essentially just does an escort mission while the other 4 members of their team are actually playing.
To me, Push was long, snowbally, imbalanced, and had a contradictory design. As I said, I can’t comment on Flashpoint or Clash because I haven’t played them, so I have no idea why people would or wouldn’t care for them. But I got really sick of Push really early, because of the aforementioned reasons.
I do wish more maps had at least the option of flank routes. The Eichenwalde choke was always so brutal, all you had to do was watch the bridge and that door to the left and you were golden. Same with Hanamura for the first point, and to some extent Volskaya (ome heroes could take the flank if they had mobility, but if the only alternative entry route is hero specific that’s not good.) Even if the flank route isn’t great, as long as there’s at least two directions an enemy can come from it makes it a lot more interesting than “keep your camera right here and just shoot what moves.”