Keep in mind what we know of Lancel.
- Tywin's line clearly considers Kevan's line to be inferior Lannisters. This is inescapable, Kevan and Lancel are continually shown (and treated) like background Lannisters.
- Lancel was effectively on the level of hired help/squire/handmaiden to Cersei. She commands him to do her bidding. As far as we're concerned, seeing things from Cersei's point of view, Lancel is a mute hired help.
It stands to reason that Lancel is unfulfilled. Though a Lannister, his opinion is never asked for, he enjoys little to no status, and he's continually made to feel inferior by his own family.
This is very similar to how Theon feels about the Starks, after returning to the Iron Islands and before being captured by the Boltons. He's "the lesser Stark", and what does he do when he gets an ounce of power from his father? He immediately wishes to prove himself worthy and actively oppose the family that made him feel lesser. Lancel is doing exactly the same thing.
The High Sparrow's agenda, in essence, promises that "the meek shall inherit the Earth". This catches Lancel's attention. The promise of the underdogs becoming victorious is obviously alluring to someone who feels like an underdog.
He has no grudge against rich people or the king's rule
Not as far as we know, but we also know little of him. Given how we see him treated, it's more than likely that he's searching for purpose more than anything else.
Note that he's one of the High Sparrow's zealots, he's clearly trying to fulfill his purpose to the best of his abilities.
In fact, after Tyrion was declared traitor and kingslayer, Lancel had chances of being "Lord of Casterly Rock" as implied by Jaime.
- Lancel may not have expected this, due to being perpetually ignored.
- I'm pretty sure Kevan would be Lord before Lancel (similar to how Stannis were to initially inherit the throne from Robert, not Shireen).
- Jaime already made a deal with Tywin to become Lord of Casterly Rock in exchange for Tyrion's survival. Even if the deal did not go through, it sets a precedent that Tywin will still attempt to give Jaime the lordship (and thus take it away from Lancel).
What happens to your name? Who carries the lion banner into future battles? Your nephews? Lancel Lannister?
Jaime says this in a derogatory manner. This both offends Lancel (if he would hear it) while also indirectly stating that Tywin would never want Lancel to be Lord of Casterly Rock.
Why Jamie mentioned Lancel and not Kevan is not clear. Maybe simply because they both consider Lancel as an even less suited Lord than Kevan, and Jaime was trying to make a point.
We can see Lancel was more like his cousin Jaime
Jaime is a renowned swordsman and knight. While his kingslaying reputation earns him no favors, no one is refuting his swordmanship or capabilities as a leader (as he leads the Kingsguard).
Lancel is a wine-bearing helper. They are nothing alike.
- He was squire to Robert and most probably was spying on him for Cersei.
That doesn't prove anything about his worth. He's not a particularly cunning spy. The only reason he's able to be Robert's squire is because Robert thinks he's doing "useless Lancel" a favor. Lancel is a spy because of his family name, not because of anything noteworthy about himself.
- He was lustful as he sleeps with his own cousin and queen.
Based on the interactions, it would be more correct to say that Cersei used Lancel as a substitute in cases where Jaime was not available.
- He was shown fighting in the Battle of the Blackwater.
That doesn't qualify him for lordship. As is always the case, many men were fighting in the battle against their will. At best, it shows that he's not a complete wimp who cowers at the first sign of trouble.
Regardless of Lancel's abilities (which I do think you're vastly overestimating), you need to consider the near definite fact that Tywin and his children consider Lancel (and Kevan) as ineffective and lesser Lannisters. Unless Tywin's entire lineage is destroyed (which may yet happen), it's unlikely that Lancel would ever hold a seat of power - unless he happened to marry into a different seat altogether.