Naturalisation - Income proof

Ask the authorities what they accept and what not.
I doubt that you would find (m)any people in the same situation on this forum or elsewhere.

In general they of course want to see one's income tax returns and information about their employment.  Certainly one’s assets and savings can influence a decision but it is an individual decision from a particular office. Someone else could theoretically have exactly the same situation in another city and get a different result. So nobody can possibly give you a definite answer except the officials you are dealing with. So just ask them.

But honestly, choosing to give up a job and be self-employed while in the process of naturalization is a huge mistake. Financially they want to know one will not need assistance and doing this takes away their security. Most new businesses fail. Free-lancers might be good at their job but not have the contacts to establish a proper customer base to survive. Even if one has saved a lot, it won’t last forever if they lose money. And the prospects of one making it as self-employed vary wildly according to the field, experience, competition etc. You probably should have waited until you had finished with naturalization before giving up a job because afterwards nobody has anything to say about it.

And one should know that it is often difficult to get an appointment to take the German test, then one waits for the official results. I did it at the Volkshochschule and the slots were given months in advance. So one has to wait until the opportunity to sign up for the following periods. And often the slots will then be filled on the first day. One needs to be there the first possible day to sign up, and try to be first in line. I had waited at least a month to be able to sign up for the next slot 3 months later. I got there in the late morning and all of the slots were already gone. Had to wait another month or two but showed up at least an hour before the office even opened.

And once all the documentation is in, it can still be months or easily a year before things get decided. And who knows how much the pandemic might slow down the process further. One should not be too impatient and constantly bother the officials about getting the answer in the end but if they somehow don’t have a document they expect to have then they don’t necessarily tell you; they wait. My naturalization took about 18 months. I was falsely told that the results of my German test would be sent directly to the office so they would know it is legitimate. But it turns out that one has to make a copy and go to the office, show the original and give them the copy. This might have lost me around 5 months’ time.

The thing is also that one might assume they do everything like the background check in Germany and the home country as to if there are any criminal charges or suspicion of anti-democratic activity already. The final document like the German test is given in and then can quickly finish the process. Wrong! They wait until every document is in before starting to do anything at all. The former approach would save time – but that is not their priority. Waiting for all documents saves them other efforts that might be for nothing if not all of the required documents show up. After all, not everyone passes the German or citizenship tests. One would assume that people would not apply for citizenship at all if they didn’t fulfill the requirements. But at least half of applicants are refused.

Thanks for the detailed reply.
My case is a long story, a bit unfortunate. I actually started the naturalisation process more than a year ago.  The only problem was that I could not show them the B1 certification at that time (which was the only outstanding supporting document). 
I actually successfully passed both B1 and citizenship tests 7 years ago, got a letter of confirmation from BAMF, and thought that was all I should get from them.  At that time, with the results, I got what I wanted: a successful application for permanent residence.
Only when I decided to go for naturalisation last year did I realize that TELC  did not send me the B1 certifcate, and the local naturalisation authority insisted on having such and ignored the confirmation letter from BAMF.  TELC could not reprint the certificate for me, because they only keep their records for 5/6 years max.
So, my naturalisation process could not proceed further at the time when I still had a full-time job. 

Now, after these months of corona, I have a chance to take the B1 test soon, only that I have just quit my job (under an agreement with my former company long time ago).  If the authority refuses to proceed my application later due to no employment income,  well, I guess I will have to attempt later, when I have some income again. 

I am actually thinking about getting rental income by investing in a small apartment.  Do you know if the authority has a certain income threshold for the naturalisation application?

The general rule is that one should be earning at least enough that they don't need any kind of public assistance.