should your CV include unfunded grant applications?
No.
Definitely list pending grant proposals. But rejected grant proposals should fall into the memory hole.
Also, do not list your rejected journal submissions, the awards you applied for but did not win, or the graduate schools that did not admit you.
When applying to a TT position at a research intensive school, the search committee is going to want to know want your first research project is going to be and who would likely fund it. A failed grant application along with an explanation can go a ways towards providing that information. One line on a CV probably isn't helpful and I would not include failed applications on my CV for such a position. I would however mention one of the proposal in my cover letter. Being able to say that you have a proposal and have identified potential funders (which comes first is a little bit of a chicken and egg thing) is a good thing. Being able to say you applied to funding scheme X and while you didn't get funded got useful feedback and you are now revising the application for funding scheme Y is even better. Obviously tailor this if the funding scheme gave a scores/percentiles/stage and/or allows you to resubmit.
You should ask yourself, who will read my CV and what do I want to achieve by tailoring it.
In your case the CV is part of an application for a position in academia, it should make an impression on the people in the search committee. From my experience people usually list only funded proposals in their CV. So including unfunded proposals will be considered as something odd. I don't say this is strictly forbidden, but you should have an extremely good reason for doing this. And I can't think of any. An unfunded proposal will be (from my experience in committees) not considered as an achievement. On the contrary, if you have a lot of funded proposals you might want to include the unfunded, too show your "success rate". But I wouldn't even recommend this.
If you want to prove that you can write about prospective future research, you should write an excellent research statement. You could address your experiences in this place.
Again, their is no right or wrong. So this is my very personal opinion.