There are some sites I'd like to ask for a link which I think I have an excellent chance of getting, but I'm sure it would end up on the "Links" page along with a lot of others. Is that beneficial at all for SEO?
There are some sites I'd like to ask for a link which I think I have an excellent chance of getting
Your chances may not be as excellent as you think. Out of all the link requests I've gotten over the years, I can only recall one that I accommodated. Usually the other site is worthless. Nowadays I seldom even look at them, nor do I bother to take the time to reply. I've gotten leery of clicking links in emails from unknown persons anyway.
If you do make a link request, I suggest that you don't start out by telling them what a fantastic site they have, or how your site is the perfect resource for their readers. Also, don't make up a story about how you stumbled on their fantastic site while trying to help some disabled kids with their homework, or some such.
I've gone through several periods where I've used link "arrangements" and then using automated linking programs without providing a link back. Probably a waste of your time, it certainly was for me.
netmeg says links need to make sense, and I believe that to be very true. If you want a link to pass value, an independent viewer should be able to determine why a link points to A, which links to B, which links to C.