I would definitely not cut the runners, that's how it spreads and produces more fruit.

I agree let them run. Unless they are planted in pots and you aren't keeping the runners. The first year you want your strawberries working on putting down good roots so pick any ripe fruit and pinch off any remaining flowers or small green strawberries. Next year you will have much healthier plants producing tasty fruit.

Let me make myself clearer. As I said, I will propagate the runners, meaning I will let them take root and become a plant. However, the roots of the runners have already become brown. Will they still take root?

Also, once I'm done propagating them, making about 4 or 5 of them, and then cuut them, if I cut off it rest of the future runners and fruit, will the mother plant still grow its roots stronger?

Yes but you only have to do this the first year.

It is its first year. I didn't even get to eat its first fruit because a bird ate it

Hey that was me last year! Now I'm eating tasty fruit and thinking about making jam with the leftovers. You've got the right idea, it'll be worth it next year, just remember to protect with bird netting early next spring.