This is because he rivalry draws its strength from the fact that both young women see something in the other they hate about themselves. While it's clear to us that Betty is willing to use every advantage she can to make the case of "I saw him first, he's mine," she doesn't especially want to see it that way because it collides messily with her self-image of passive victim of an aggressor who'll pull every dirty trick in the book to get what she wants. Similarly, Veronica isn't especially fond of the idea that she's actually the same sort of love-sick, needy, clingy fool hung up on someone who's pretty much as replaceable as everyone else on the planet that she sees Betty as being because she's got it in her head that she's smarter than that.
Given the company's insistence on not resolving the issue because the target audience wants to see a funny, funny hot mess filled with screwed-up people racing around in circles to maintain an untenable and unhealthy situation, it's kind of obvious that the poor man's Andy Hardy is pretty much the template for every passive male lead in a harem manga because his weakness and flawed personality keeps the tension on. His inability to simply make a clean break because he overestimates how catastrophic the results of resolute action will be not only make things worse for everyone else, it further angers the ultimate loser of all losers exceeding all others: Reggie.