There are two reasons crit chance feels... Egh.
There's actually a considerable amount of items (roughly 10) that give it as a stat. Only one class of champions can really use this stat at all reasonably (with a few outliers), and that's Marksmen. Most every other stat can be used reasonably by at least two other classes of champion.
Defensive stats are mainly used on tanks and bruisers.
Raw attack damage is used well by assassins, bruisers and marksmen.
Ability power is loved by a bunch of mages, and some supports, or "enchanters".
And pretty much everybody loves CDR.
Resistance penetration is really a multiplier to your damage, but Marksmen are unique in that they have resistance penetration, as well as a whole other stat to multiply their damage. Crit chance.
Because it essentially acts as a multiplier to their overall damage, which gets a little more efficient the more raw damage you have, it's almost able to be formulaic.
There are a couple ways I really liked how others handled it, but they aren't really applicable to League.
DotA 2 has three items that critical chance. Six people have it as part of their kit. As passives. But, stacking critical hit has diminishing returns, since they're all independent modifiers, with different chances that provide different multipliers. It's not really good to buy more than one. Or any, if you have it innately.
In Dawngate (RIP), you actually couldn't purchase crit chance directly. There was also no such thing as ability power and attack damage. There were two unified stats instead: Power and Mastery.
One point of Power increased a right-clickers' auto attack damage more than it would a mage's. But overall mages had higher Power scaling on their abilities.
One point of Mastery both increased critical strike chance by 1%, but also increased your ability damage by 0.5%.
It was a stat that multiple classes could use. Dawngate had an incredible stat system and I loved it. There was no such thing as dead items on people (a la AP Riven), instead, certain things just didn't work as well, but they still worked.
In League, well, certain things are just stupid but you go for it, AP-Tear Zed. Don't let anyone dull your sparkle. Just don't expect to be in good terms with your team.
League has Randuin's Omen and Ninja Tabi, to keep crit right-clickers somewhat in check. They hurt, but there are ways to hurt their damage output. It's not perfect but at least it's something.
If it changed to your suggested Damage Amp, at least we wouldn't have right-clickers fishing for a lucky crit to put them at an advantage before even considering committing! :V