According to the question, John M. Taurek says that giving everyone an equal chance of survival seems most fair/reasonable to him. This seems like something you might not agree with. You might decide instead that you don't like killing people regardless of how unfair it may be that only 1 person dies and the other 5 survive.

In order to understand why your understanding of the solution suggested by John M. Taurek seems wrong, we can consider what other implications may arise from deciding that giving everyone an equal chance of survival is most fair/reasonable.
In the following situations, John M. Taurek represents a person who wants to give everyone an equal chance of survival above all else.

Consider a similar choice between 0 and 6 people killed. In this case, John would still be happy with the coin flip, as everyone has an equal 50% chance to die.
You may not be happy with this case as there is an equally fair option of just killing 0 people and saving all 6. This seems better for everyone involved, but without some definition of utility, you would not be able to say that.

In fact, it gets worse than that, as John M. Taurek would also be equally happy with the fair option of just killing all 6 if he was given the option. He might even prefer it if he expected that living people would be treated less fairly than dead people in the future.

It gets worse again if we consider a trolley with 1 person before a junction who is always killed, and 5 people after the junction who could be killed if John pulls a lever. In this case, John will decide to kill the extra 5 people in the interest of fairness.

All of these cases being worse requires you to have some reason to prefer living people to dead people. If you don't prefer living people to dead people, then your understanding of John M. Taurek's solution is one you should be happy with and you shouldn't argue against it.

In the 1 vs 5 case, you could appeal to ethical egoism instead of utilitarianism and claim that you don't like killing people, and would prefer to kill 1 than 5, and therefore should do that.