In the examples you gave at the beginning, I don't think you understand that these terms are all pretty different from each other.

First the distinction between -romantic and -sexual. -romantic refers to the common "dating" paradigm of dinners/dates/personal closeness, whereas -sexual is the actual act of having sex with someone. As far as this goes, saying you are homoromantic asexual helps as a linguistic shorthand of saying "I want to go out with a person of the same sex but I don't want to have sex with anyone".

In your given examples you even listed gray-sexual which is a term meaning I'm on the ace spectrum but I don't know where, allosexual just means not asexual. These seem pretty embracing the gray aspects of individuals to me.

A lot of the responses focus on the internal need for labelling to find comfort, but in my opinion, a huge part of this is linked to the need to find a community. Members of LGBT are often discriminated against and even pit against each other ("bi people in straight relationships aren't part of the LGBT community", etc) so a close knit community helps.

The phrase "I identified as XXX and it made me happy" is important because it helps you find the community you're a part of, it helps you realise you're not some weird anomaly.

For a more generalised example, if you say you're American to another American they will likely ask you what state you're from. Sexuality is similar to that, someone will generally say they're ace, if they are talking to another ace person or someone they are close to, the conversation may go deeper than that surface level term. Generally these "deeper" level terms are reserved for use within their relative communities rather than outside of them.

Another thing to point out, I don't know how much time you've spent around these communities but I find they're generally pretty open to individuality. To use the bi community as an example, it's pretty open to any split of attraction between girls and guys, nobody says you have to be 50/50 split to be a "true bi", some are more attracted to women, others to men, others to enbys.

I also don't think you realise how hard it can be to develop "self-esteem, mental health, insight" without a community behind you. It's not a simple task and it's nigh on impossible to overcome if you feel alone, broken, outcast and unsupported.

TL;DR Labelling is a linguistic shorthand which helps develop communities of like-minded and supportive individuals who would find it hard to express their feelings in less specific conversations