The BBC was set up to be advertising free, so that option is not a part of the current structure.
The license fee was established because of fundamental beliefs about issues like free riding, externalities and more. You might prefer a subscription based model - I'm sort of on the fence myself, but it's not obviously wrong - but the BBC license fee was set up out of an explicit disbelief that such systems would work. Granted, some of the issues were technological - you couldn't actually stop people watching OTA broadcasts at the time. But even though those have changed, the beliefs about the funding structure have not.
The BBC does have some advertising on it, if you can call it that. Most of it tends to be inhouse. So in addition to TV programmes, in the past I have seen them advertising "Radio Times" (a magazine they used to own giving TV listings), tie-in books, TV licences, DVDs/VHS, and their other channels and digital services. They also cross-promote their material. When David Tennant was playing Doctor Who, you would frequently see BBC News 24 being featured in the programme.
Nowhere near as bad as other channels in that sense, but still there.
Historically, there have been also been substantial numbers of people who watched the BBC without licences in the Republic of Ireland and the Netherlands, but they couldn't do a thing about it. The BBC tended to be watched in the east of the Republic of Ireland and near the border with Northern Ireland. (Not so much in France from what I can tell.) Many of the houses in Dublin used to have massive tall TV aerials to receive it. Most have been removed now. (Within the Republic of Ireland, RTÉ is funded by their own licence system, but also has proper advertising on it, unlike the BBC. It has had similar questions about it.)
Well, I can. I'm old enough to remember "Radio Times", and other magazines, being advertised quite openly on BBC1 and 2 as well as their radio stations. I think they had to sell their share in "Radio Times", after government pressure, but they still do many tie-in books. (Especially true of their science fiction franchises such as "Doctor Who", which has dozens of official books based on it.)
The other advertising includes heavy promotion of BBC linked charities such as Comic Relief, Children in Need and so on. These charities make big money and there have been some questions about how that money is used and where.
BBC advertising is less obnoxious than commercial channels, but it is still there. In addition, the BBC owns BBC America (which carries commercial breaks), as well as having shares in services accessible in the UK including the "BritBox" streaming service, and the digital channels "Dave" and "UK Gold" which all have normal commercial breaks.
1) I suspect that I am older than you, but either way, probably the same cohort.
2) I have a very hard time considering a media organization mentioning its own products and activities in its content as "advertising". If you want to use the word that way, be my guest, but my understanding (and I think most people's understanding) of the term implies a 3rd party paying a media organization to include marketing content in their output.
3) Fair point about BBC America, but I don't think it really invalidates the point.
The BBC does not carry advertising in the same way as ITV, but a certain amount of content qualifies. I don't include trailers, but I do include promotion of their own non-TV products, the TV licence, promotion of the corporation as a whole (the BBC has done a number of nostalgia reels and songs — their cover of "Perfect Day" years ago would qualify.) and so on.
The BBC has a perpetual Catch 22 around self-advertising, much like the NHS.
The license fee was established because of fundamental beliefs about issues like free riding, externalities and more. You might prefer a subscription based model - I'm sort of on the fence myself, but it's not obviously wrong - but the BBC license fee was set up out of an explicit disbelief that such systems would work. Granted, some of the issues were technological - you couldn't actually stop people watching OTA broadcasts at the time. But even though those have changed, the beliefs about the funding structure have not.