tl;dr - This account is relatively new, and I've been chat restricted twice. Each time only a single game was in the reform card, where I said fuck words and got flagged by the banbot.
I'd like to see the reform cards, 'cause there's obviously more here than you're letting on. The IFS won't ban you simply for saying "fuck" - or swearing in general. It's when you start using those swears to attack others that it becomes a problem.
If you've already cleared both Chat Restrictions, your only method of getting access to the chatlogs from your previous reform cards would be to contact Riot Support and ask for them to dig them up.
As for them being single matches - you'd have to be pretty inflammatory to merit a punishment from a single match, so either you were just that, or you've a history of generally negative behavior.
From what I gather, the honor climb back to being eligible to earn keys and boxes again is pretty insurmountable...
It's only insurmountable if you maintain dishonorable/negative behavior. If you actually show positive, sportsmanlike behavior, it's not that bad of a reclimb.
...and seeing as I'm positive that receiving an infraction shines a spotlight on your account to Riot...
That's weird phrasing, but no. The only thing that "shines a spotlight" on your account is your behavior. Behave poorly, and you get punished. Behave well, and you don't. Simple as that.
...I'm wondering if it's just simpler to make a new account and relevel; or should I just stick it out on this account and never talk in game again?
That depends. Are you going to change your behavior if you start up a new account? Or are you just going to make the same mistakes, get punished again, and wind up right back here at square one?
...because I'm convinced that the inmates are running the asylum with the way the report system works...
You'd be wrong.
...and I'll be damned if my money is going to be stolen because a bunch of kids think I should be banned for saying mean words on the internet.
Well, Riot's not a bunch of kids, and your money wouldn't be stolen. You would have given it willingly.
Do remember that you agreed to the ToU, which state;
5.1. Can I troll, flame, threaten or harass people while using the Riot Services? (No. If you do, you might get banned.)
While using the Riot Services, you must comply with all laws, rules and regulations in the jurisdiction in which you reside. You must also comply with certain additional rules that govern your use of the Riot Services (the “Code of Conduct”). The Code of Conduct is not meant to be exhaustive, and we reserve the right to modify it at any time, as well as take appropriate disciplinary measures including account termination and deletion to protect the integrity and spirit of the Riot Services, regardless of whether a specific behavior is listed in the policy as inappropriate. In addition to the Code of Conduct, please review the Summoner’s Code for additional guidance on exemplary gameplay behavior.
So, if what you mean by "saying mean words" is "flaming people", then, yeah, you'll get punished for flaming people. Since we don't have your chat logs, that's all that can be assumed.