Lyxen
Great Old One
That's the big one
I'm sorry, but it's not really big. All it says is that "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. ". And it's not even in the main text, it's in a side box, and uses words like "most" and "usually" (or even "around", what does it mean, within 5 feet ? 10 feet ? 20 feet ? Certainly not 50 feet even less 100 feet), so these are neither rules nor a certainty.
Another is from surprise where if you aren't trying to be stealthy, you are noticed.
No, this is not what is said. It is said that if you are not trying to be stealthy, you will probably not achieve surprise on your opponent. Don't read too much into this."It does not mean that you will notice them, and it certainly does not mean that you will even see them or hear them. RAW, there is no proof of this.
You are generally aware of what goes on around you in combat. That would include a wizard chanting, waving his fingers and thumbing bat poo.
No, once more, this is your personal inference and it is certainly not in the rules, and if it is prove it. I am pretty sure that you will fail.
You would probably notice him if he was right in front of you, but what if he was 20 feet away ? 50 feet away ? 100 feet away ? Behind an orc swinging his axe and thirsting for your blood ? Behind a couple of orcs ? Behind 20 orcs ? Behind 100 orcs ? And what about lighting ? What if the light was poor ? Or if he was in a completely obscure area but still seeing you standing in the light ? And what about ambiant noise ? What guarantee do you have that you can even hear the chanting ? What if the orcs above are screaming war cries at you ? Would you still hear the wizard ? And by the way how loud is the chanting ? Nothing is said here, but I can certainly chant with nothing more than a humming noise since the words are not the source of the spell's power. Same with the somatc gestures, how discrete can they be, since you need only one hand free ?
There are no rules there, only purely the DM's (local) ruling and certainly no right to any entitlement of a player to noticing spell casting. None.
By the way, I'm not saying that you are wrong playing it that way, just pointing out that this is YOUR. LOCAL. RULING. Not the RAW. And that someone, like me, who still gives a lot of value to the Subtle Spell Metamagic power (my halfling sorceress took this one as her first choice), but who rules that you will not perceive a wizard casting 20 feet away from behind orcs charging at you is equally right if you look at the RAW.
Subtle spell is critical for social situations which are usually in a quiet environment, but in the middle of combat, you will find zero proof in the RAW that it is noticeable.
Without the caster being stealthy in some way(subtle spell, invisibility, etc.), you're going to notice a caster.
Do you have proof of this RAW ? I'm certain that you don't.
If you read the components section on page 203 of the PHB, they are pretty darn noticeable.
On the contrary, as demonstrated above, nothing says the chanting has to be specifically loud, and the gestures need only one hand.
I have been in hundreds of LARP skirmishes with between a couple to hundreds of combattants, and I can absolutely guarantee that, even in LARPs where there is no actual magic (sad, but it's a sad real world that we live in

Once more, your interpretation is right at your table, but please stop trying to convince us that the rules are written that way, it's simply not true, which is a good thing because it makes other more mysterious and equally interesting interpretations just as valid.
Obviously if there's dim light(moonlight), etc., that would possibly obscure thing. But again if you read the components section, spells being cast are really obvious.
Again no, they are not, demonstrated above.