As both defeat cosmic forces regularly, either. Although the FF and JSA are good bets as well as are the Xmen and everyone else because unless I'm missing something, they always save the world!
JLA. Last decade Avengers has generally had a fairly low-power level, though made up for by having multiple teams.
The JLA has had a couple leagues in the last decade, some better than others, but the Dick-lead league was good, the pre-Infinite Crisis league was very good, and while the post-IC league accomplished little it did have heavy hitters.
Nope, neither have, but expressing a preference for one or the other, I'll go for the one that does so more often and is moderately better equipped to do so, even if both are fully capable of the job.
The Avengers of the last decade haven't fought a lot of cosmic threats, I'll note. Most of the cosmic stuff has been off fighting the space enemies.
I'd want a team that's willing to do ANYTHING to save human lives, including KILLING. I don't know if that's still a big issue or not with the Justice League and their No-killing policy.
But when it comes to high end cosmic threats, the JSA's the team to get. While the JLA was usually dealing with Amazo, The General, Prometheus, the JSA is throwing down with the King of Tears, Extant with Warlogog, 5d imps, old god Gog.. The JLA does big threats too, but less regularly than the JSA (And they had the JSA with them for Crisis Times Five, and other teams for Imperiex and Megaddon..)
How is it ludicrous to have a preference when they've been dealing with different numbers of threats?
Just because neither has failed and both are good doesn't mean that there's no difference whatsoever or criteria on which to hold a preference. That's the ludicrous position.
Cosmic threats in the last decade that come to mind. Scarlet Witch (dealt with by alt-universe heroes). Annihilus (Nova and co). Phalanx (ditto). Secret Invasion (not just/primarily Avengers). War of Kings (space types again). Cancerverse (ditto). Sentry/Void (Avengers). Fear Itself (Avengers).
Now, they have no failures, but some were really tough for them, and for the most part they've been dealing far more with non-cosmic threats. In the last decade they've far more focused on stuff like Civil War, and Norman Osborn and his dark reign, and so on and so forth, and similarly the teams are built that way- it's only now during Heroic Age that they're going beyond one herald-level big hitter on the team.
The JLA has faced multiple Crisis events in that time, and a number of other cosmic threats. Stuff like universal truth itself unraveling (JLA Golden Perfect), or a White Martian invasion, or 6th dimension beings (the story where the JLA was split into secret IDs and hero selves), or Obsidian Age, or the whole Final Crisis thing, or Gaia itself thinking of wiping out humanity (JLA #100). They and the JLE sub-team fought and got rid of the Whorlogog controlled by Manchester Black.
The Mark Waid and Joe Kelly runs of JLA were in the last decade, and in those the JLA fought cosmic threats pretty often.