If you have a good on-topic question, you should be posting it to the main site for the benefit of others who may have the same question, if not to get multiple answers and insights which you may not have considered.
If you just ask it in chat (and "leave the issue as is"), very few users will see it and it will be essentially impossible to find by searching.
The intention of chat is most definitely not to take good questions away from the main site.
If you're not sure about the on-topicness of the question, your best bet is to search / ask about the category of the question on Meta - again trying to maximise the number of people benefiting from a post.
As for using chat to improve your questions: unlike the main site, all messages are equal, and other things may be discussed, so any given message can quickly get lost in the noise / responses and others can't edit what you post (apart from copy-paste edits, which are clunky).
That leads me to conclude that the interface is not particularly well suited to anything more complex than a one-sentence question like:
Would this question be appropriate - ...?
How do I best phrase this - ...?
But these also don't lend themselves to that much improvement - posts are often rewritten entirely to make them more appropriate, and a whole lot more detail may be necessary to really understand what you're trying to ask.
It's usually better to pay attention to actions taken on your questions (such as edits, comments, downvotes and close votes), to seek to understand and learn from these actions and try to ask better questions in future.
So, yes, if you want to do a QUICK, BRIEF sanity check before posting, you can probably do so in chat.
Quick as in it shouldn't really be an extended back-and-forth to get to figure out whether it's appropriate. Brief as in the actual message should be short.
And of course chat would also make sense for a question that lends itself better to discussion and/or opinion (which would be off topic for the main site).
But don't post your actual on-topic questions as is in chat (without explicitly asking for improvement) - that will lead to answers more than suggested improvements and you'll just be demotivated from posting your question on the main site.
Also, try not to overdo it - if your messages of the above makes up a significant portion of all messages, you should probably tone it down.