but don't pretend that without the EU I would have been able to have an experience like that because it is a lie, I couldn't.
Well that's odd because plenty of students, artists etc did what you did before the EU ever existed, and two of my adult children have travelled, worked and lived in exactly the way you describe in the last year or so, in countries that aren't even in Europe, never mind in the EU. 
Okay so they went with a bit more than 100 quid on them, but they were still gap year kids on a tight budget, picking up casual work where they could.
And plenty of countries require you to apply for a visa in person. In fact, the UK does for some non-EU nationals. And some people in the UK have to apply in person for a US visa, they can't use the ESTA programme because eg they have a criminal record (which could be a very minor offence) or because they've been to a country the US does not like (eg Syria).
Yes I know, but what does that have to do with what we are talking about in the context of leaving the EU?
We aren't talking about what we require of other nationalities outside the EU before UK visas are issued, we are talking about what other countries require of us to be allowed there.
If there are grounds for concern and further investigation, such as the circumstances you mentioned, then visas won't be automatically granted for certain countries, the US being the most obvious example, but as a general rule visas for the vast majority of places are not especially difficult to obtain for British citizens, whether you have to appear in person at the embassy or not. And as I said, for visas to countries outside of the EU our membership of it is totally irrelevant.
How do you imagine things will change for British people travelling outside of Europe, when we are no longer in the EU? Do you think for visa purposes immigration officials in the USA or Thailand or South Africa or Iran even know or give much of a stuff about what federal clubs we do or don't belong to? A British passport is a British passport.