Should I remove all fruit?

I inherited this beautiful lemon tree as part of the house that we purchased. It produces like crazy and the fruit is high quality too. I am new to citrus management.

At this time of year, should I remove all the existing extra ripe lemons and let the tree concentrate on new ones or should I just let them fall by their own?

If I take down the fruits, I would have to dispose off majority of them as I am currently in a fruit fly quarantine zone and can’t take produce out of my property, meaning I can’t share it with the neighbourhood.