Actually, Cait's early game is considerably weaker than you seem to notice, I don't know if you ever find yourself in the bot lane (maybe you're a jungler or support main? i dunno, it doesn't matter >_>)
Lets break it down a little: there are two ways to go, Q max (piltover peacemaker) and W max (yordle snap trap).
Q max gives you: Waveclear, quicker farming/shoving, some long range harass.
W max gives you: the ability to get traps down in cool places, the ability to set up defensive and offensive hot zones.
So, if you max w, all the way up until level 9, you have zero waveclear, cannot shove into tower to get traps under your turret, and are vulnerable to any sort of long range trading (jihn e, twitch w+e, trist eqw combo, Ashe w, MF Q, Jinx w, varus q, draven's whole kit, lucian q, corki q and r, ez q and r, Jayce q+e or q, graves q (though why you would ever play graves bot lane now is beyond me), kallista q and e, Kogmaw w, and probably many others that I'm forgetting). All of the champs from the aforementioned previous ellipses have considerable waveclear, or have the option to build considerable waveclear: that means that cait, with only traps, will never make it to the turret in a good matchup because these champs will ALL shove the wave harder than she can. harder shove means more minions lost to turret aggro, and by the first back (~ 5 - 7 minutes in) you can have enough cs to effectively zone a cait and force her to farm under turret.
If, on the other hand, you max q, your lane presence up until 9 is ok, your traps have a 30 second cooldown and you can only place three of them, and you aren't super vulnerable to trades from enemy adcs. You also have the added benefit of not losing any and all lane presence by the first back because you can get creeps with your q (you can't get creeps with the traps).
I'd like to suggest that if you're ever in a position where cait has shoved up to tower and has lots of traps down, that it is probably close to lvl 14, maybe 15 or 20 minutes into the game, and cait taking your tower isn't oppressive, its the only thing her kit excels at. OR, you just really really suck at being an ADC, don't understand lane presence at all, and a locked in to a recalcitrant playstyle.
So, if you find yourself in a position where people are playing w rush cait, and you happen to blow at ADC (not saying you do personally, just talking about if people in league are bad generally), the entire playing field around her backside is super open. Lets presume that your jungler blows too. You have the option to walk up around her backside, and give her some real hell. that will cause her to reposition any available traps she has, you can flash or dash over a wall to get away from her, and then you go back to tower with her traps now in a new position. If she builds a Riot enforced Meta ADC build (IE or ER rush) she doesn't get significant cdr until 2 or 3 items into the build. That means those newly repositioned traps are gonna stay for a while at the early levels.
that also means she can't trap to protect from jungler or mid ganks (since 4 manning in the bot lane seems to be the most popular option >_>), and if she does get ganked, she has no damage.
Cait doesn't really get super strong until maybe level 11 or 12, basically mid game (and much later if you go w rush). Her early game loses hard to almost everyone (but not vayne) if she does trap max. her early game is a skill matchup (meaning the more skilled person at their champ wins) if she maxes q, but that's average for ADC champions (that is to say: not OP). her late game can be impressive, but she gets outranged or out dps'd by just about every hypercarry in the game. She's like one of the most balanced ADC's, i think (her weak ult not included in that assessment: her ult is about 1/6th weaker at level 6, 11, and 16 than every other adc with a comparable ult, presumably, I guess, to make up for the longer range) in that she doesn't have many serious weaknesses after around level 11 or 12, but she also lacks significant strengths compared to many of her counterparts. She's like Mario in Super Smash brothers, you know? not many weaknesses, but not many strengths either.
SOOOOOO....
now that I've built a huge ass wall of analytical text no one is ever going to read, I'll break it down in the terms you mentioned above:
you said 'In laning phase she puts traps under your turret unless you push HARD to their tower, which really isn't that easy considering her waveclear early on. But the biggest problem with her is the fact that she is a lane bully, who never falls off.'
In laning phase she puts traps under your turret unless you push hard, which isn't easy: that's patently false in two places; in laning phase, every ADC out waveclears her (vayne excpeted) unless she maxes q first: so 1) its easy to waveclear if she's putting down traps, and 2) if she has lots of waveclear AND traps, it is no longer laning phase; Traps can go under turret, but they have a considerable cooldown, and can only be placed in a group of 3 until lvl 12 (not enough traps to effectively block your movement).
To address you second point, of her being a lane bully who never falls off, you might be technically correct there. but she also never picks up, in the way that a vayne, lucian, kogmaw, tristana, twitch, ashe, jhin, kallista or draven does, either. All of those champions are far stronger in the late game that Caitlyn is. If she can withstand her considerably weak early game, she can maintain relevant DPS into the late game, yes. But she lacks significant trading potential versus any burst lane, and that shouldn't be overlooked. she also gets out dps'd early game by a significant champion pool, too.
