Courgette question

One of my courgette plants appears to have broken off in the centre (ie where the main flower stalk grows from) so presumably it won't flower anymore.  If that's the case can I dig it out?

My courgette/pumpkin patch is like an unexplored jungle and I almost need a machete to get through it so if I can get one past-it plant out it'll make fighting my way through the jungle a bit easier.

On the same note can I take out some of the lower leaves on plants that are still active?  My daughters keep complaining that by the time I find the courgettes in the middle of the jungle they're so big they could almost be hollowed out and used as boats!

One of mine did this, think it was a plain Zucchini type, but rather than bin it, I potted it on into a flower bucket and popped it in the greenhouse to see if I can get a couple of late croppers.

A b it of an experiment, but it's fun to see which plants growing seasons you can extend by a bit of protective cropping.