I would put it in much the same way you've written it here.
You mentioned that your job is stressful and demanding and you're not growing career wise. Make your direct report think about what's keeping you at the company and what reason you have to stay. Perhaps they think that you're comfortable with what you're doing and you don't really mind that you haven't been given anything extra, perhaps they're just thinking that you want what everyone else is getting?
Below's a few talking points for you.
Be extremely clear about what you want and where you want to end up. Most people don't want to stay in the same job for the rest of their lives, so let your manager know what you aspire to, what's driving you. Show that you're keen to learn.
Don't forget to tie it into a benefit to the company, why should they put you through the training? How does that turn into a benefit for the business and how will they get a return on their investment? Will they end up losing the "invaluable person" they have in their current role? Talk about how you could up-skill/cross-skill someone else into your role.
If you're determined to reach a goal and there's a roadblock in the way, ask "How can we overcome that?". If it's something that simply cannot be overcome, then you really don't have much option other than to look for another job that will provide what you're after.
On a side note
It's best not to let things slide for so long, if you're determined to get something, letting it go by for 7 years without having a path to your destination doesn't exactly scream determination. You have very little reason to show loyalty to a company that can't keep it's promises. At the end of the day you need to look out for yourself. "You're #1" as someone else here put it!