Character development clearly falls under character building and is off-topic.
I disagree with JDługosz on that "if some good questions could use it, then why not".
The first items listed as off-topic in the help center are:
- Actions of individual characters, rather than elements of the world they inhabit
- Character building
"But if there are some good questions where the said tag could be appropriate..."
If a question is on-topic but happen to contain an element which can be used for character development, then they do not need the tag "character-development". It's a bad idea to add that tag to those questions, even if they are good questions. Just because there are good questions where such tag could be added does not mean that it's a good tag; it will only open up the door for bad questions which are, according to the rules, off-topic. Adding a tag for something which is off-topic is to beg for trouble, people will not know if they should VtC a question with such tag for being off-topic or not, and those who get their questions closed will not understand why.
"But I really need help with my character development"
Well, then you got three options:
Phrase the question so that it's not blatantly off-topic. If you can cut away all unnecessary parts of the question and ask for a single element of how you want your character to be, but phrasing it so that it's not obviously about character development, then you can still ask the question (but without the tag). I admit, this is not an easy feat and I cannot give any good examples of how to do it but if you can phrase the question about something on-topic, then what you actually use it for later is irrelevant.
Start lobbying for making character development on-topic. Things can be changed, even the scope of the site, if you get enough amount of the users on your side. I must warn you, though, it's not going to be easy to change what is on-topic. The rules were set during the creation of this site and character development was chosen as off topic as it has no clear answer, it's mainly going to be opinion based answers (which is bad as good answers should be objectively good). So the chance that you will succeed is "slim to none", but as it is a site made for and by the users, if enough users want something, then it can be implemented.
Ask somewhere else. I'm not suggesting that you leave WB.SE, but for those questions which are not possible to answer here, you will need to seek help elsewhere.
In my opinion, the best way to go is simply to follow the first point - try your best to rephrase your question to be on topic, that way others can get useful tips from the answers generated.
Edit - only marginally relevant stuff below
The following part is not relevant to the question "do we want the character development tag?", but I add it anyway as I thought of some aspects regarding character building.
I did think of one example of how to make a character development question kinda on-topic:
This is perhaps not a good example, but I add it in order to offer some help of how to ask such question. Lets say that you want your character to go from naïve and happy person to a more grim and hardened person, then you cannot really ask "how to make my character go grim and hardened" "what type of event will make my character go evil" as that would not only be off-topic for being about character development, it would be off-topic as way too broad.
Instead, think through what your character will go through in the story and ask something concrete from there. As example, you could ask "what happen, in general, to persons going through a brutal war, where they see someone close to them get butchered". That question can be answered subjectively (but might still be off-topic) by looking at how real people behave after surviving a brutal event and would give answers such as "they risk developing post traumtaic stress disorders". If you then read up on PTSDs, you can find out how a person with it will react in various situations.
A thought on character development - proper research before asking questions is the key:
The best way to build a character is to write about what you know. If you, e.g., have no clue about how it is to live in Japan, then maybe it is a bad idea to write a story about life in Japan without first going there yourself for a while. An alternative is to not write about Japan, but a completely fictional culture inspired by from what you know about Japan. However, you can also research Japanese culture a lot first, without the need to live there, and then ask specific questions about aspects of the culture which you want to use.
If you, on the other hand, is an author whom is too shy to ever ask another person out for a date, then it will be hard to write about love and romance, but not impossible. Internet provides ample of opportunities for a shy, introvert, person to find someone else to chat with and experience some kind of relationship building. Sure, it will not be the exact same thing as going on an actual date with someone, but it will at least be a base to start from (and might be a way to overcome the social awkwardness and meet someone IRL). Maybe such author should not write about how it is to be an extrovert Casanova, but rather how it is to be a shy, yet romantic, person whom gradually gathers the courage to approach their true love. My point with this is that, again, background research is the key to good character building and it is possible to research love, even for a shy introvert person (and, to be fully honest, even persons whom have a lot of experience of love can be extremely bad at writing about it).