does the router broadcast a signal that the phone can pick up through its own data connection?
No. The gain antenna (Yagi in this case) needs to be connected to the 4g phone or dongle which works on the band on which your wireless internet connection is operating - if in fact that is the case.
The term wireless internet can mean two things in current usage. It's essential that the correct application is understood. One interpretation is that it uses one of the mobile phone bands and the nearest cells to operate to either a mobile phone or 3/4 enabled tablet or laptop for internet usage. I think this appears to be the intention for the antenna under offer and it is assumed that it will be feeding a device which can utilize the 3G or 4G signal to provide the signal for the internet connection.
The second interpretation is where people understand wifi (or a networked connection which includes internet access) from a router whose internet connection comes though copper or fibre to be "wireless internet'. Strictly speaking this is not wireless internet at all - but often people think it is.
I think more specific information needed about your present internet equipment and connection before a definitive answer can be given. At present, based on what I think I read, it isn't going to assist..
However, it's midnight here - I've just come home from a party and I'm tired. This might be more clear in the cold light of day.