The US has public radio and television without advertising as well, but instead of licensing it is funded through personal and corporate donations, together with government grants coming out of the general budget.
On a few programs you might get the host of the showing saying at the end mentioning one of the sponsors, but that's the worst it gets. The website (for radio, I don't see any on the TV website)) does have a few non-awful display ads, but streaming everything is free.
In my personal opinion this is a more progressive model - poor people don't have to pay money for public television access.
It's more progressive, yes, but when you look at the quality of output that the BBC produces the two really aren't comparable.
In the UK, I think we take the BBC for granted. With a fairly-limited budget they produce extremely high-quality and varied ad-free programming, news and have an excellent web presence. So many shows are exported and shown across the world, it can only be a benefit for the country to have our culture, and values projected in such a way.
Additionally, my two favourite stations are BBC4 and Radio 4. BBC4 shows a large number of niche documentaries that simply wouldn't ever be considered, let alone funded, shot and aired for other channels, and Radio 4 is simply unparalleled in quality.
Other (mostly European) countries rely on a tax to fund or partially fund state TV/radio so that the government doesn't broadcast government propaganda over state TV or control it by cutting their budget. It's also easier for them to make cultural shows as opposed to Oprah style reality TV that's the equivalent of clickbait and sells ads. In some of these countries state run TV is the most propaganda/advertising free TV. BBC is one example.
NPR and PBS neither air reality TV type content, or are outlets for government propaganda.
I understand that's one stated reason for the license tax, but the American model pretty well proves that it isn't necessary to get independent content that's not ad-driven.
You seem to forget that the US is about 5x the population of the UK.
Voluntary donations are nice but assuming generosity in the US and UK being the same, the UK would end up with 5x less funds than what PBS and NPR have (about US$550 and $270 yearly, respectively).
The BBC has about US$6.5B and produces a much wider array of programmes than PBS and NPR combined. So not relying on public's mandatory contribution would clearly not work.
I don't know how it works in the US and whlile I fully agree with you about NPR (not familiar with PBS) I also know that it wouldn't work in my Eastern European country (an EU and NATO member). Our gov't tried to subvert national TV several times for their own corrupt schemes - with parliamentary backing. They just got rid of the TV tax, everybody and their dog voted for them and now they're indtrducing new and even more interesting taxes in place of the ones they got rid of to get the votes. Some people have a really short attention span.
In the UK, this also means we don't have any advertising on the BBC channels or website, which is nice.
https://en.wikipedia.org/wiki/Television_licence