Oh, you're right about there is no obstacle, and when the idea of blowing up the bridge come up, it was cut by the idiot, and never addressed.
He, Kannahibi, didn't suggest destroying the bridges, he suggested
charging though the bridges to the other side (the other officer was who suggested it), which even with numbers is suicidal and dumb.
Destroying bridges is a double-edged sword and a last resort for a number of reasons. Yes you might destroy routes of access and gain significant range, but the enemy also gains the same benefits and if their side have longed-ranged equipment like snipers, artillery or vehicles, you just played yourself especially if you blow them up early. Worse even, if the river isn't that deep and/or their vehicles can wade up to a certain depth, destroying them just gives them a field with plenty of cover for an approach.
And that's before the logistical and political issues they rise. Bridges have been VERY important in and out of war since millennia ago, and for good reasons. You don't destroy them nilly-willy.
If later there's a stupid reason for letting that bridge up, the soldiers are not going to be happy to that.
The very reason I could think of blowing them up is that the enemy has tanks/IFVs and they're going to cross with them. But even then, IF they choose to cross they would potentially face building after building of soldiers with anti-armor weapons. So if they have the tanks, it would be better to use their superior range to shell the opposite riverbank from safety and send the infantry later than risk a panzerfaust to the face at point-blank.
And I think their CO suspect this is what's going to happen. That's why they keep the bridges whole (and the poor sods on the first line being called expendables...), to lure them where they want to and to temp them to cross.
Defending a bridge like Benkei, and The Berserker of Stamford Bridge is a romanticized story, young man. Search for your medals in school sport festival.
It depends on what they're defending it with, what the enemy is sending against you, and the era. Sure, when you only have to fear iron blades and pointy sticks, defending a bridge with a few men is feasible with the proper tactics and individual power, but when it's tanks or heavy cavalry what they send...