When do you accept friendship from a romantic interest?

When a guy dates a girl for a little while, and then gets faced with rejection and the ultimate "Let's be friends" or "Can we still be friends" it is considered an act of self respect to decline the offer and walk away from the girl for good. But personally, I can't help feel that rejecting a girl's offer of friendship is giving up on what could end up being a strong, solid friendship. Not to mention the romantic opportunities you may find in her social circle. Sure you don't want to look at someone for their potential, but rather for who they are then and there.

But i'm battling to understand this concept of walking away from the friendship just because she said no to something more intimate. I do understand that accepting the "friendzone" or walking away should be an act of accepting reality for what it is, and put all thoughts of romance with that person out of your mind and treat her as a friend, just like she ulitmately would with you. And of course, move on to other romantic prospects. I just don't understand why you'd say to a girl who friendzones you "My interest in you is romantic not platonic, thus i do not want your friendship"

Some guys go into friendships with the hopes that one day he'll get into her pants and does everything he can for her with this direction in mind. But Here i'm talking about genuinely accepting the reality of the friendzone, and moving onwards to bigger and better fish.

My question is this: How do you decide whether to walk away from someone who friendzones you, or genuinely accept their friendship offer? And under which contexts would you do either?
When do you accept friendship from a romantic interest?
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