That router does not have the feature you need to even attempt this. The router needs what is called client-bridge mode or maybe repeater mode.
You likely have 2 other major issues.
First to run in this mode you need a feature called WDS enabled. Most public wifi will not have that feature turned on. Some tplink routers can run in what is called universal repeater mode but you have to read the specs to see if it does not all model have the ability even if they can run repeater mode.
Next and the more likely thing that will prevent it is almost nobody runs a open wifi anymore. Way to easy for bad people to use it with no ability to track them. If you look at the way say a coffee shop or hotel that has open wifi you most times need to at least click on agreement to some web page or maybe enter a code to allow it. A router has no ability to enter data into a web page.