Hiya!
Which is totally fine. Some folks do if it doesn't help them run a better game that they want to run.
The problem that I see is that, each individual player or table may find alignment very useful, but it's so depending on that individual's or table's belief that it's very hard for it to be consistent across the board.
No. I tried to get this across upthread: the individual is NOT the determining factor of what is "lawful" or "evil" or whatever; that's defined in the Rule Books.
PLEASE NOTE: I USE THE 1e AD&D ALIGNMENT DESCRIPTIONS! The ones in the 5e PHB are...uh... less than useful; I find them either confusing or outright wrong. So, going forward, that's my basis.
Here's the description from 1e DMG, page 23:
"Law And Chaos: The opposition here is between organized groups and
individuals. That is, law dictates that order and organization is necessary
and desirable, while chaos holds to the opposite view. Law generally
supports the group as more important than the individual, while chaos
promotes the individual over the group".
"Good And Evil: Basically stated, the tenets of good are human rights, or in
the case of AD&D, creature rights. Each creature is entitled to life, relative
freedom, and the prospect of happiness. Cruelty and suffering are undesirable.
Evil, on the other hand, does not concern itself with rights or
happiness; purpose is the determinant".
With those definitions in place, we can continue...
Like, would a Lawful person respect a high-ranking individual who is incompetent?
Yes.
(Or who isn't evil, good, or neutral enough.)
Irrelevant if we are ignoring G/N/E.
Wanting to remove that individual from power and put a better-suited person in their place is at least a bit chaotic, especially if there aren't any hard and fast laws for expelling someone from office.
Not even a bit chaotic. An incompetent person would likely put the group/society/organization/etc at risk...this is 'bad', so needs to be fixed. If there were no rules/laws for expelling someone...then the Lawful person would go through the proper means of getting a rule/law ratified that would then allow this.
And a Chaotic person can have great respect for someone just because of their title and status, even if there's no actual power in that title. They just might decide that the highest-ranked member of the Pastry Chef Guild is more worthy of respect than the Queen, because the Master Pastry Chef is really, really good at their job and clearly earned that title through hard work, while the Queen just had to be born to the right family.
Yes, but that has nothing to do with, well, anything other than the Chaotic personal feelings/belief. I fail to see how this is somehow an example of something a Chaotic person "shouldn't do".
And that doesn't even take into consideration people who play Lawful Awful or Chaotic Stupid or Neutral Boring.
Problems with people a the table...not the Alignment system.
Basically, while yes, I'm sure that you find alignment useful, I just don't think it's as useful as people think it is. I think people come in with a lot of their own decisions as to how individuals or societies work and only afterwards take alignment into consideration. (Seriously, how many people really play elven societies as Chaotic Good?)
I play Elven societies as CG.

The usefulness of the Alignment system is for the DM to get an nice, big, birds eye view of "the gist of the character/monster". I mean, if you present two DM's with a bad guy. You tell one DM "Nellmastur is trying to take over the city. He's Lawful Evil", and you tell the other DM the description of ... "Nellmastur is trying to take over the city. He's Chaotic Evil". ...and that is ALL you have to go on, what's going to happen?
The DM will start thinking of his methods, personality and all that other good stuff of HOW he would go about trying to achieve his goal of taking over the city. After the campaign, you could talk to the Players and DM's and find that both had, in all likelihood, quite different campaign experiences! Why? "LE" and "CE" are descriptions that intone a general "personality method of life".
That's why Alignment is useful to me. If it's the other way around... I have a full page or two of description about Nellmastur, his desires, history, likes, dislikes, methods, etc...but I do not have his Alignment...I can then give him one because I just read all about him. It doesn't matter if Nellmastur firmly believes that him taking over the city is "for the greater Good and will improve the lives of everyone!"...if he tries to accomplish those goals by killing, torturing, public executions, draconian laws that change based on his word alone, confiscation of all wealth to be distributed as he sees fit, and forced conscription of all citizens between 13 and 50 years of age....he is NOT going to be "Lawful Good". The rules of the multiverse don't care about your particular feelings or views on what is/isn't "right or wrong". (see my copy/pasted 1e DMG bits above...

).
^_^
Paul L. Ming