sigafooNHL: Salary cap added in 2004.
NBA: Soft "salary" cap added in 1984.
MLB: Luxury tax added in 1997.
Soccer is the is the closer comparison since, to my knowledge, no European leagues have any sort of caps on spending. Which is why you also have fairly predictable local leagues. For instance in the NFL, the last 12 seasons have had 11 different teams win the championship. Whereas if you want to find the last 7 different winners of the English Premier League (since Premier is only 20 teams compared the NFL's 32 teams), you'd have to go back almost 30 years to find 7 different winners of the premier league.
But if you look at the UEFA Champions Cup, see it being broken up a bit more. A bit harder to maintain year after year winning success with teams who are able to spend similiar amounts money as you on talent. Plus, there is too much top-tier talent to contain onto one team.
These are interesting if slightly naive comparisons. Competitiveness is the headline PR reason for salary caps, but the far more influential reason is for owners to control costs. It's also worth nothing that dynasty type teams of the past like the 49ers also cheated on the financial rules of the day to assemble their teams, as PSG have done recently in European soccer. Although things like draft systems make it clear that there is a design in the sports to bring losing teams up to a competitive level, it's worth remembering that when it comes to money there are always other interests in play.
The comparison of NFL winners (for example) vs EPL winners has a number of causes. The length of the league in the EPL is far longer, 38 games vs 19-20 games for the Superbowl winner so the quality has more chances to even out random events. The NFL uses a playoff system - the EPL is won in league points accrued, knockout rounds make the tournament much less predictable. And also factors such as the draft, salary rules, etc, help to even out the teams quality, but they are far from the biggest part.
In terms of a given game soccer is an enormously random sport. If you look at the difference between rich and poor teams it's extraordinary that the top teams dominate as little as they do. As you've noted the Champions League tournament, dominated by a knockout format, is much less predictable. The headline EPL example that you've skirted around mentioning of course is that Leicester City won the league with a player budget less than 15% of Manchester City's.
As far as TF2 goes, it's a very predictable game. Players have gravitated towards 5cp that by design softens the impact that random events or unexpected spikes in skill can have. It rewards sustained consistent high level play. Unlocks that create random events are banned. Players prize always having the feeling of the potential to have counter-play. The games mechanics don't really allow for the snowballing of advantages in the same way as something like Counterstrike, and very high skill one shot kills are rare so it's hard for big swings in momentum to happen. Protecting the uber is basically all we have, and in 5cp that's often not even enough for a round by itself. Sometimes not even a point.
The alternative modes aren't much better. Payload has a slow moving cart and is probably worse than 5cp in this regard, nobody plays CTF (although this does have quite a high potential snowballing element but without the mechanics to back it up), AD has been shunned amongst other things because it's too random which probably makes it a pretty good candidate for these purposes, KOTH seems to have an endless dearth of acceptable maps, although the potential for snipers to run rampant has made it potentially more entertaining.
Use of weirdo unlocks obviously hasn't derailed Froyotech in 7s so they're probably not a game changer.
Which leaves us mostly with format changes. Fewer rounds, shorter matches, knockout games, unfamiliar maps, etc. Seeing as Froyo seem to be able to win games 5-0 at will messing with the structure might have negligible effect, and would other teams scrim oddball maps Froyo wouldn't? Maybe. As someone else has mentioned handicapping may be an option.
There are a lot of options to explore here if breaking up Froyotech's hegemony is the goal, and a structural solution is more likely to be of lasting benefit. Regardless, it's been a pretty effective way to generate free publicity and make the league a talked about subject even if it's mostly criticism.