I came across a video recently posted on youtube of my former jujutsu teacher, John Bear Shihan demonstrating futari kata in 1989. Was surprised to see that I was the main receiver, joined partway through the video by Neil Phillips.
At that time, I was a shodan (1st Dan black belt) and Neil was probably 3rd kyu brown belt. This kata was a training exercise for 5th kyu purple belt level.
Mountains and rivers 
What a great find. Could you have stopped him from throwing you had you wanted to? Wonderful to see you in action.
Maybe. We did practice and learn techniques for countering throws. Involves good timing. But in a situation where the attack is committed and the response also, its usually too dangerous to try to counter the throw in the training situation, unless pre-arranged as practice. Reason being that most of the throws involve a strike or lock, or even the throw itself becomes lethal if the person is not devoting their attention to receiving the technique safely. So the counter move may succeed, and if it does not there is a high risk of serious injury or worse. In real life of course, that risk may be worth taking as the consequence of not countering may have 100% probability of bad outcome. What we tended to do instead was to accept the throw and know how to receive it without damage, so we could bounce back and take the initiative.
I can give 3 examples that can happen when attempting to counter a technique during training.
1. John Bear was demonstrating a block and throw in response to a front kick to stomach. I thought to myself if I kick faster I’ll get him and I tried it. He realized the kick was coming in fast and blocked it very fast resulting in a massive haematoma on my shin. He didn’t need to throw me either, as I fell over from the strike.
2. We were teaching a beginner a technique for using a headlock to throw the opponent (the beginner). He said to Neil that he could easily counter the technique. So Neil did it again and after getting the beginner in the headlock struck him very hard in the solar plexus with his other fist before throwing someone whose breathing had stopped. The beginner had not realized there was a solar plexus strike involved in the technique.
3. We were training with another club not familiar with our techniques. John Bear went into the headlock and throw I mentioned above and his opponent threw himself into a judo-style side breakfall. He would have broken his own neck if John had not sensed what he was doing and released the headlock (which otherwise would have prevented his neck turning with his body into the side breakfall).