Problem is, BBC (and to a lesser extent RTE) are not these unbiased services simply presenting facts. Their staff have certain views on political issues and they present stories in a way that is biased to those views. They effectively tell their viewers the range of acceptable opinions they are supposed to have.
A lot of their entertainment shows are junk too and I do not see why they should be funded through a license fee - if there is a demand for them, they could be funded privately. The BBC website is full of tabloid-like news - that is unacceptable.
If we are to have publicly funded media, it should be boring, i.e. serious. Host 2-hour long in depth debates on highly controversial issues between the most prominent people that oppose each other, carefully moderated (arguably it would actually be better to do this in print rather than tv format). Do cultural shows that promote "high culture" that people otherwise might not have engaged with. Counter the natural tendencies to look up to sports and entertainment personalities with a focus on scientists and engineers.
Having spent plenty of time in the USA, I can't fathom how a system with so many ads and skewed "news" could be better. Why would you trust an unelected private corporation with no motivation or mandate for transparency more than a non profit org that is at least in theory answerable to the people? If nothing else it provides a common reference to reality that allows a real conversation to happen. The lack of this has proven in the last few months to be a real problem.
Most people don't question paying for socialised defence, education, roads etc... many services they may not use personally on a daily basis, because they understand the value of having a cohesive society as a whole.