The word invisible does actually not imply anything magical or supernatural in natural language, it means "not bisible" and can mean quite mundane hiddeness. Genre expectations in Fantasy do tens to Amp thst up, I suppose.
Not bisible! I know where I've gone wrong now!
5e.2014 (PHB p177) had the following advice: "In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen." Again, not automatic but not impossible.
Yeah, and if the 2024 rules had that text, I'd feel a lot better. They don't. And it's a subject that warranted further discussion in the 2024 DMG.
The fact is that there are systems - like Pathfinder 2 - that have better text covering it. Here's the PF2 Hide skill:
###
HIDE (PF2)
You huddle behind
cover or greater cover or deeper into concealment to become
hidden, rather than
observed. The GM rolls your
Stealth check in secret and compares the result to the Perception DC of each creature you're observed by but that you have cover or greater cover against or are concealed from. You get a +2 circumstance bonus to your check if you have standard cover (or +4 from greater cover).
Success If the creature could see you, you're now hidden from it instead of observed. If you were hidden from or
undetected by the creature, you retain that condition.
If you successfully become hidden to a creature but then cease to have cover or greater cover against it or be
concealed from it, you become observed again. You cease being hidden if you do anything except Hide,
Sneak, or Step. If you attempt to Strike a creature, the creature remains
off-guard against that attack, and you then become observed. If you do anything else, you become observed just before you act unless the GM determines otherwise. The GM might allow you to perform a particularly unobtrusive action without being noticed, possibly requiring another Stealth check.
If a creature uses
Seek to make you observed by it, you must successfully Hide to become hidden from it again.
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Now, PF2 has really gone deep with the rules (different states of observed, hidden and undetected) and those aren't really necessary for 5E (much though I like them), but there's a lot of grey in 5E about how a DM should resolve it. Which, when you move from table to table, can cause disconnect.