Yes, it is a war that can't be won
What I did in my garden (subtropical asia) is to engage a mercenary ! I searched and found local weeds that are useful to me and introduced them in the garden. With my help the "good weeds" battled and replaced the "bad weeds". Now I just "whack and harvest".
The weeds I choose are Gotu kola (Centella asiatica), lemon grass for the sunny areas. I choose a local fern for the shadow areas, I don't use the fern other than marvel at it's beauty.
I definitely abandoned the idea to have a perfect lawn as it is just unhealthy for the local eco-system.
Oh and the lemon grass clippings are great mulch for the banana