After a capital raise, companies who’ve sold securities, aka issuers, are required by securities law to track and report according to the rules associated with the exemption they used to raise.
Most US corporations outsource the transfer agent/registrar function. They do this due to the scale of items to manage, continually changing regulations, and the ability to achieve separation from conflicts of interest.
Transfer agents also have a duty to enforce the eligibility requirements. For example, many securities have a ‘hold’ period before they become ‘seasoned’ — just one example of a factor of eligibility.