It is not true, and it is the virus being manufactured. The idea is that the virus manufactured slowly enough that your body can learn to fight it.
You ARE correct that the virus is not confined to the injection site.
Once your body has successfully fought off any virus, you are immune to that virus for life. It doesn't prevent infection, but it mitigates the effects of such infection since your body can fight it off much faster than before.
For Covid19, it's rather pointless, since this virus does not kill, and is instead the mildest variant of the Covid/SARS series so far discovered.
It's just the DEMOCRATS getting all twisted up about nothing (again). In this case, the purpose was to try to make Trump's successes look bad and to justify their election fraud.
This fear mongering by DEMOCRATS started the current economic depression. It DESTROYED quite a few businesses, and they are GONE. They are not coming back.