Only a small number of all public sector jobs would would require a degree.
I think you may be underestimating the number of public service jobs that need a degree. For example, the public service employees huge numbers of teachers, and they all require a degree. Then there are other groups who obviously require a degree, but are in smaller numbers, such as doctors, lawyers, lecturers, etc.
And of course these days all newly-qualified nurses have degrees. I don't know whether they are making full use of all their medical training that they get during the nursing degree. I would hope that they are now doing a lot of jobs that were previously done by doctors. But you won't easily find a newly-qualified nurse without a degree.
There are also jobs that don't necessarily require a degree, but you may struggle to find a suitably qualified person who doesn't have one. For example, most accountants have a degree these days, even if you don't have to have a degree to qualify as an accountant. You might be able to run the Office of the Comptroller and Auditor General, or the National Treasury Management Agency without any accountants, but you probably don't want to.
Similarly, you probably want some civil engineers in the National Roads Authority, some statisticians in the CSO, some economists in the ESRI, some scientists in Science Foundation Ireland, and at least someone with a degree in the Higher Education Authority. Maybe some of these bodies should be shut down, but as long as you keep them open you will need specialist staff with formal qualifications.