It is standard that all grants go through the institution rather than directly to the investigator. The only exceptions I can think of involve direct reimbursement of costs like travel expenses that are paid by the individual for a business purpose, not research grants.
Remember that you are using a bunch of institutional resources that cost money: some examples from your post would include the time of the review board, the cost to license the software you use, other resources like your office and computer and the utility bills for electricity and heat or air conditioning, etc. The institution may allow their employees to use these things "for free"/without an explicit cost to the user, but they expect to cover those expenses within their internal accounting from a portion of all external grants received, regardless of the size of the grant.
It's important to follow your institution's requirements on this sort of thing and seek their guidance, even if it's a small amount, unless there is an explicit exemption granted to you. If you don't, you're likely to experience exactly the penalties/consequences you have experienced.
I didn't have to declare it but I did anyway
You almost certainly are required to declare grant awards to your employer. You are also likely required to document received funds for tax and regulatory purposes.
Is it not possible for a researcher to pay for the participants privately?
Acceptable sources of funding are up to the board that reviews your research, but it's possible that the transfer or appearance of transfer of funds directly from a researcher to a participant could create ethical problems where the research participants may feel a direct debt to the researcher. There may also be tax and other concerns. Money from international sources may create further complications.
You need to use the professional regulatory resources at your institution to help you conduct research according to all regulations. Even if the potential issues are relatively minor, it's normally up to an independent institutional committee to make these decisions; it's not safe to let researchers make their own decisions about research ethics.
Receiving funds creates a conflict of interest situation: the educational and research interests of your institution come into conflict with your interest in receiving future funds and the funding source's interests (which are unclear without more information). A conflict of interest does not mean that wrongdoing has occurred, but it is necessary to avoid even the appearance of wrongdoing and to mitigate any concerns that could cause harm to research participants or to the reputation of the institution. Any secrecy/lack of clear disclosure increases the likelihood that others will perceive a higher risk of damaging conflicts of interest.
It doesn't sound like you've been fired from your position, so that suggests the institution would still like to work with you: you'll need to get guidance from your department and your institution's research support people on what your next steps are to come into compliance. You're going to have to trust and obey them. Continuing to make decisions based on what you guess or assume is correct is likely to cause you more harm and could include losing your position which might also impact your immigration/visa status.