No shit.
This is exactly how the real world works. Don't try to pretend that Pina's country is to Russia as Japan is to the US. They got stomped in their first full incursion, and obliterated when Japan retaliated. Guess who that's like? Countries/forces in the Middle East.
The US doesn't pay for hostages. At best they trade them (which the JSDF in this series has been doing). But if they are engaged in operations in the region, they simply take them back by force. All the time.
If the Taliban had an American (or Japanese, or British, or Australian, or random NGO) in captivity, the US would attack the Taliban, assault the compound, take the hostage, and kill everyone inside.
Stop pretending this is a fight between 1st and 2nd World powers. It's a 1st world nation attacking a backwater 3rd world nation. And their magic doesn't mean shit. They tried it already with dragons and shit. Artillery and anti-aircraft weapons tore through them. Mages would get killed via sniper fire, the series has already shown Japan's sniper teams to be rather competent.