The choice how to enter formulas is a matter of taste. For example, since I have years of experience using LaTeX, by far the easiest way for me is to enter the formula in MathJax. I remember some users here on meta mentioning AsciiMath, I can imagine that it would be a rather fast way to enter a formula for somebody who has enough experience with that format. And, naturally, if somebody has no experience with MathJax/LaTeX, it might take some time to learn it and such user would probably prefer something similar to CodeCogs editor or some other formula editor.1 (Various editors often include some unnecessary stuff - the MathJax code could be written in a cleaner way or shortened - but I do not consider this a big problem.)
For people who want to have some kind of editor accessible directly when editing on the site, maybe installing some kind of userscript would be an option. But I did not find a userscript which would do exactly this - and creating a userscript which would have full functionality of formula editor would be rather difficult if not impossible, probably something like buttons with common MathJax commands would be doable.2
As already mentioned in comments, there were some older suggestions to incorporate some kind of MathJax editor directly into the built-in editor on Stack Exchange. (For example, WYSIWYG for math formulas? or How to increase service/traffic, a set of suggestions.) In my opinion, this is unlikely to happen.3 (And I am not really sure whether this would be a good idea.)
A better solution to me seems to let know the users (especially beginners) that they can use some other site or application to write formulas, if they prefer to enter mathematics in some way other than directly writing the MathJax code.4 However, I'd prefer a link to some post here on meta summarizing various WYSIWYG-like ways to enter the formulas and include the use to this post. (In this way, the information does not have to restricted to one specific editor. Moreover, in case some editor stops working, a post on meta can be updated by regular users, while a link in the edit window can be only edited by a Stack Exchange employee. The same is true for editing the help center, with a few exceptions which can be edited by the mods.)
Proposal. So this would be a version of your suggestion to "bring the feature to MSE somehow" that I would support:
- We (as a community) would create a post which explains the possibility of using external tools rather than directly typing MathJax code. And post links to such tools in the answers. (I think that starting a new post might be preferable to modifying one of the existing posts5, since in this way we could clearly explain the purpose of the post. Information on some editors is mentioned here on meta, but it is scattered in various posts.)
- Links to this post could be included in the MathJax tutorial. (Since it is the place where new users are most often sent in connection with editing math.)
- I was considering also adding the link in the "helpful links" on the right next to editing window.6 But it is probably better to include a link to the MathJax tutorial rather than have two links, one to the tutorial and one to the post with editors. (If having two links rather than one is, for some reason, preferable, we could consider including it into the information for the first time users. It contains the link to the MathJax tutorial, see this post for a screenshot.)
1Several such editors are mentioned in the Wikipedia article formula editor.
Some of them even have a separate article on WP, see the Category:Formula editors. If we want to create a FAQ post about such editors, link to the Wikipedia article probably should be included. I will point out that apart from editors where you can edit a single formula at the time, there are also various editors which might be useful if somebody wants to prepare the whole post in some editor rather than directly here on the site. Several such editors are mentioned here: MathJax: better way to prepare a Math.StackExchange question?
2I found this post on Stack Apps: Math buttons for sites without MathJax support. I have not tried the script - so I cannot confirm whether it still works. And it is unlikely that it will still be working after Stack Exchange switches to the new editor. You can see some screenshots in the linked post. However, this script is intended for sites without MathJax - as explained in the post, it is for entering some common math symbols as HTML.
3Very recently there was this post: Opt-in alpha test for a new Stacks editor. (It is still shown in the community bulletin among the posts featured on Meta Stack Exchange.) I'd say that this post gives you an idea which modifications of the Stack Exchange editors are currently considered.
4Even if using various formula editors might be easier for new users, I still think that knowing MathJax/LaTeX is useful for somebody who often uses this site. For example, if you want to subsequently edit your post, it is much easier to edit the source than entering the formula again from scratch. It is also useful for searching. For example, you might enter MathJax formula in the built-in search on this site or into Google. While various search engines for math (see: How to search on this site?) usually have some kind of editor for entering math formulas, if there is an option to enter directly LaTeX, it offers the users more control on the search query.
5I was able to find these ones: Latex editor suggestion and WYSIWYG for math formulas?. On the main site: Free software to create equations and export to various formats.
6Here is a screenshot showing what I see on the right after I click on Ask Question. I was a bit surprised that the link to the MathJax tutorial isn't included in the section name "Helpful Links". (I thought that it was added there when this change was implemented: Show “how to ask” advice before a new user asks a question. But it seems that I misremembered it and this link was only included into the info shown to the first-time askers.)