Its simple, you bring your neighbour's enemy to your country. Your neighbour want that enemy to go away. He makes concessions with you, you throw the enemy out of your country.I am skeptical. How will Philipine use US base as a leverage for negotiating with China? First it is an insult to supply geopolitical enemy of China in their plan of encirclement. Second there is nothing they can do to kick US out even if China makes a good counter offer.
Here is an equally absurd idea. How about Mexico be a forward supply base for China and Russia? That will surely 'strengthen Mexico's hand negotiating with US' right?
They can kick Americans out in the end but only after expending tremendous political capital, Philippines is not Japan (yet)
If you ask, why can't China emulate what the US can do to a neighbour hosting its enemy, its because China still doesn't have the same hard power as the US has