Is it possible to bi-directionally route a 4G signal using a connected set of antennas only?
No this will not work for a very good and easy to understand reason. The Yagi and the patch antennas, like any other antenna, have their specific efficiencies and associated losses. So your system proposed based on the seller's claims, assumes there are no losses going from the Yagi to the patch. (By losses I mean a reduction in electrical signal due to conductor resistances and other imperfections in the cables etc.) Then it is also assumed that efficiency to capture and to transmit radio signals somehow, are perfect 100%. This is never the case, even for some of the best antennas your money can buy ! There are also cable losses and potential impedance mismatches that will render the system useless without overcoming with amplifier gain ! You would need bi-directional amplifiers to get something out of this, a lot more costly and beyond the scope of what the buyer would want for the low price ... Don't waste your money ...