Point 3) Moving and grappling is incredibly hard, and requires more then rocking up and having some general atheticism - try talking to some actual grapplers and suggesting that a runner (ranks in athetlics) could take any of them down without any specialised training. Even better, go do some actual grappling and see how little crossover you get from any existing atheticism.
Point 4) There is no need to be an asshole here.
Easy there guys. Let's ratchet down the tension a little bit -- let me see if I can restate things here to make sure we're talking productively rather than gearing up for a flamewar.
Let's look at this from a Game Effect vs Narrative Effect perspective for a moment.
Let's start with what we have as written:
1 Full Action • Attack Action
The character attempts to shove an opponent in a straight line.
He moves up to his Speed directly toward 1 opponent whose Size
may be up to 1 category bigger than his own. When he enters the
opponent’s square, the characters make an opposed Athletics check.
If the bullrusher is mounted or operating a personal vehicle he uses
the Ride skill instead. The bigger combatant gains a +2 bonus with this
check per Size category of difference between them.
If the bullrusher loses this opposed check, he moves back 5 ft. and becomes
sprawled; otherwise, his opponent is pushed directly back 5 ft. + an additional
5 ft. per 4 points of difference between the results, after which he becomes
sprawled. If this path is obstructed, [special stuff] occurs.
Game Effect = Defender Push + Defender Sprawled (Takedown)
Limited by: 1x Attacker's Speed, Size Differential
On Failure: Attacker Sprawled
Possible Narrative Effects = success "A great hit drives the runningback back and puts him on the ground" failure "Look at that broken tackle"
1 Half Action • Attack Action
The character attempts to trip an adjacent opponent whose
Size may be no larger than 1 category bigger than his own.
The characters make an opposed Acrobatics check. The bigger
character gains a +2 bonus per Size category of difference. If the
character wins, the opponent becomes sprawled; otherwise, the
tripping character becomes flat-footed.
Game Effect: Defender Sprawled (Takedown)
Limited by: Size Differential
On Failure: Attacker Flat-Footed (Vulnerable but still on his feet)
Possible Narrative Effects = success "The running back gets tripped up at the line" failure " look at the runningback hurdle those outstretched arms trying to trip him."
Continuing the football comparison, Grappling is mostly what happens in the trenches
Open Stance (Stance): Once per round when you have 2
hands free and an opponent misses you with a melee or unarmed
attack, you may immediately Grapple or Trip him as a free action.
You may not take move actions (though you may still take 5-ft.
Bonus Steps as normal).
Could be construed to be what Olinemen do on most plays when a pass rusher's first move is unsuccessful.
Speaking first to
Point 2) To "chase someone down and put them on the ground" you only need to make either make a move and then trip action or a bullrush action. That's it. The game has that well covered, no need to go any further there.
This I agree with. This is a basic takedown. However, there's definitely room to add some specialization, but maybe not enough for a full feat.
A Bullrush-specialization Trick might give a character the ability to Move 2X speed in the Bullrush and on success may choose to become Prone and inflict his unarmed damage in addition to the Bullrush effect?
Point 1) "Any noob with athletics" categorically cannot move quadrouple their speed and then initiate a grapple (so yes, that particular use costs 2 feats - obviously).
Point 3) Moving and grappling is incredibly hard
Yup. Again, continuing the NFL analogy, on a Kickoff, you see the Special Teams essentially sprinting half the field (pretty close to that 4x Speed movement) and then trying to take the ballcarrier down. To me: Hitting him high feels like a Bullrush, hitting him low and grabbing the legs feels like a Trip. When the ballcarrier gets stood-up and mobbed, with the defenders trying to rip the ball away? That sounds like a Grapple.
Rather than write a new feat to cover this, what about the Charge (Run Trick): You may make 1 free attack at any point during your movement (ignoring adjacency for that attack only).
If you really want to turn this into a feat, maybe go the route of allowing a Trip or Bullrush or even initiating a Grapple as a Free Attack as part of a Charge? Heck, since Charging Mastery already gives you access to 2 Free Attacks, maybe interact with that, so allow Half Action (Trip) as a Free Attack, but with 2 Attacks available, allow Bullrush or Grapple at the end of a charge?
It doesn't strike me as odd that many NFL Special Teams players might have the Charging Basics and even Charging Mastery feats...