I find it perfectly convincing...provided somebody is taking the time to look. Tracks in mud or sand or snow are pretty obvious, as is the crunching of the snow or splooshing of the mud. A "hole" in heavy rain is also quite easy to find if you look for it. Noise in otherwise-quiet surroundings also makes one easier to find, but it's rarely perfect given how easily sound tends to distort.
But noise when there's lots of other noise to mask yours? Or any of the above when in the heat of battle with something else? Much, much less likely.
It's fine to rule that way depending on your group, but that "provided..." is not a caveat you can find in the RAW itself - it doesn't mention any special time or effort devoted to knowing where the creature is above and beyond the general awareness that a character is always assumed to have about it's surroundings. The general condition is "In combat, most creatures stay alert for signs of danger all around," and in as much as that alertness can be presumed to include staying alert for unexpected noises, the general condition presents an assumption of characters basically always hearing what goes on, unless something specifically stops that from happening.
Sure, a battle's a noisy, chaotic place, and that makes some sense, but that's never pointed out to have any particular effects on any of the creatures in a fight. A battle's a noisy, chaotic place,
but also our characters are fantasy heroes who are used to battle with strange and unexpected creatures, they're not confused town guards or turnip farmers, and it's not too noisy or chaotic to hear someone whisper a spell as it's being cast (
counterspell) or to make a bard's song inaudible or to stop the PC's from having a lively conversation on the battlefield. Arguably, part of what a Dex (Stealth) check represents is your ability to
use that noise and chaos to effectively not be heard, because it's not like you can't kick a stone or step on a kobold's tail and make it yelp or whatever. Being silent on a noisy chaotic battlefield involving fantasy heroes isn't an automatic thing that can be easily assumed.
It's in the wording itself - when you
try to hide. Hiding isn't something you do without
trying.
This is how RAW points to "Invisible != undetectable," not just in certain circumstances, but as a default assumption that needs to be actively changed by something in the world.
Ultimately, though, it IS up to the DM, as always. I think there's also good gameplay reasons why a DM might
want to have invisible creatures also be detectable unless they've done something extra.
My inference here - and please correct me if I'm wrong - is that those gameplay reasons would involve making things easier for the players and-or faster overall, and those are no more than personal preferences.
You say "no more than" like there is some alternative that's more than personal preferences.
That gameplay reason also points at what the RAW's intent is, because when figuring out what 5e's intended design is when it's potentially ambiguous, you can often wager that good designers would like their game to play easier and faster, and would not like their game to turn into 15 minutes of Marco Polo any more than they want their game to turn into 15 minutes of bonus-stacking.
I suppose if someone really likes the Marco Polo-ing, that'd be a counterpoint to that reason for the RAW's intent, but is anyone really a fan of not knowing where the enemy that you're going to have to take out sooner or later is? Or of constantly tracking the "last space we saw you in"? Or of PC's easily avoiding damage thanks to not participating in the fight? Because those are some of the reasons why it's a pet peeve for me - those things are Not Fun.