I might be weird, but I find "out of date" stuff to be pretty interesting, and I'd honestly prefer to browse a collection of that than stuff that's scrupulously "up to date."
The difficulty with being the 'library of last resort' is that you need to be careful with the stock: if you have the only remaining copy it's not obviously a great idea to lend it out! In the case of the BL, their rarer stuff can only be accessed, supervised, in their reading rooms. Even the non-unique material is often only ILLed to another library's reading room rather than for lending.
I'm the same. Before I moved from the city of Brisbane (Aust) I used to often stop at this old book store on my way to uni. This is a book store that would store / sell older books and manuals, and it had a reasonably nice technical section.
I like to read older ways of doing things, and thinking back to the difference between what they knew then compared to what we know and have available now. Sometimes I get hopeful that there's some "old method" that could help solve something of today's challenges, which has just been lost in time, but that hasn't happened yet, atleast for my work. :)
(Most of the stuff there was for civil engineering, but occasionally had some nice finds. I found a 1960's radio engineering textbook that was stuffed with 1960's money including $1 and $2 bank notes, made of paper!)
Some for the content. Some for the vintage look and feel.
I thoroughly enjoy reading old magazines and such with local info. For example there was a trade publication called The Black Diamond that has all kinds of info on eastern kentucky coalfields and the towns and camps around them.
I gave my wife an old pocket book on how a wife should behave (as a joke, of course) but it truly is an interesting read to see what was considered "appropriate" 100 years ago.