That may be a comforting way to blame some group of people for your justifiable stress and worry, but it isn't supported by actual data. Take a look at Table 13 in this report. Transmission results from cases, not hospitalizations, and for whatever reasons right now in the UK where they collect a lot of detailed data vaccinated people in some age groups are twice as likely to have a COVID case as unvaccinated people.
The people spreading COVID are the vaccinated people.
You can't really tell that from this table because vaccinated people may be more likely to get a test. Anyone who thinks the whole thing is a hoax probably won't get a test until they are really sick.
Your table does show that unvaccinated people are much more likely to be hospitalized or to die because of covid.