My preference is that you usually can't really try an ability check again unless the situation somehow changes. Your result indicates your best efforts. There are some issues with the rule, but it means that the average commoner isn't guaranteed to accomplish DC 20 tasks with a bit of time. It also encourages thoughtful solutions. Try to climb the wall and fail? Go get a grappling hook and try again.
Yeah, that makes no sense to me and definitely doesn't reflect reality. Using a climbing example specifically, absolutely, I make my "best effort" each time, but here's the thing - eventually I manage to climb the rockface, even if I slipped off it the first time, or couldn't get up more than a few feet before having to hop down.
I mean, have you ever gone climbing? Serious question. Did you magically pass/fail on every rockface you encountered within 6 seconds (one round)? Obviously that's rhetorical. You didn't. Just because you make your "best effort", doesn't mean you succeed. Maybe you did something wrong? With climbing that's very easy to do, especially if you're under time-pressure.
The same applies to lockpicking - I've only done a little, super-amateur stuff (totally legal, I assure you!), but no matter how hard I try, I'm not necessarily going to succeed or fail in the first X period of time. I might get it in a few seconds, or it might take me half an hour.
There absolutely ARE situations where you only really have one shot, and time is unlikely to help. For example, trying to remember a historical fact - the chances are, either you remember it, or you don't. It's possible that you'll remember it later, but in that case I'd just put some kind of "minimum check again" limit, or just deem it "not known" and not allow another check.
I suspect what you are ignoring is the fact that in most editions of D&D, a check absolutely does
not represent "You take as long as you need to make you best efforts to achieve something", it means "You try to achieve X in Y timeframe". These are very different things. If it represented the former, your position would make more sense, but it doesn't. It represents the latter. In 3E, take 20 represents the former.
Also, if your problem is that a random weirdo will succeed on anything DC20 or less given time, you have the real answer there: RAISE THE DC ABOVE 20. It's very simple! It's not unfair. It's pretty obvious that 3E's DC tables were put together without thinking about Take 20, and if they need to be changed for that, go ahead and change them. Just don't be telling me my PC can't climb a sub-20-DC rockface, just because I failed in one 6-second period, and then claiming that makes some kind of sense, because it doesn't.