Paul asked the Corinthians to give in 2 Corinthians 8:
10 And here is my judgment about what is best for you in this matter. Last year you were the first not only to give but also to have the desire to do so. 11Now finish the work, so that your eager willingness to do it may be matched by your completion of it, according to your means. 12For if the willingness is there, the gift is acceptable according to what one has, not according to what one does not have.
13 Our desire is not that others might be relieved while you are hard pressed, but that there might be equality. 14At the present time your plenty will supply what they need, so that in turn their plenty will supply what you need. The goal is equality,
The churches were to help one another at different times.
19 What is more, he was chosen by the churches to accompany us as we carry the offering
There was no specific mention of the church of Jerusalem, but churches in general.
After he had the collection from the Corinthians, it would be under the control of Paul and others.
which we administer in order to honor the Lord himself and to show our eagerness to help. 20 We want to avoid any criticism of the way we administer this liberal gift. 21 For we are taking pains to do what is right, not only in the eyes of the Lord but also in the eyes of man.
Four chapters later,
2 Corinthians 12:
13 The only thing I failed to do, which I do in the other churches, was to become a financial burden to you. Please forgive me for this wrong!
When Paul was with them in person, he didn't need their financial support. He was financed by other churches and by his own earnings. At this point, the situation was different, the Corinthians were more financially able to give.
Why did he refuse to get financial support from the Corinthians?
He didn't. On the contrary, he was asking for their financial support in the writing of 2 Corinthians 8.
Any reason why Paul didn't collect financial support from the Corinthians earlier?
Earlier, Paul had sufficient supports from other churches. In fact, earlier they supported the Corinthian church. Different time, different measures.
The bottom line was that there would be give-and-take equality for the local churches over time.