So, if a man finds himself in a sexless marriage, at what point and with what force of will, does he decide to separate divorce and try to obtain an annulment? ...The clock is kinda ticking.
Yes, I agree. I don't know how the Church views this type of situation. I'd bet that, in orthodox times, the answer would likely be "Nothing we can do; this is a valid marriage." Annulments were extremely rare pre-V2.
From your answers above it appears that a child (or I would assume a pregnancy, even if the child is lost) lessens his case for the marriage being invalid. If children are conceived there was enough sex, even if that amounted to once per year.
I don't think marital relations are the litmus test (assuming it's happened once). The test would be related to the intention and if there were impediments to entering the marital contract. Such judgements did not happen often in times past (annulments were rare). The Church relied upon the "process" of the marriage to assume it's validity, in most cases. That is:
1. The parish priest knew both families (as most marriages were from people from the same diocese or parish).
2. If the priest didn't know one of the partners, he inquired with the other parish priest as to their reputation, history, etc.
3. Assuming all that checked out, and both parties were valid catholics (with documents to prove reception of the sacraments and schooling, etc), then the couple would proceed with marriage classes, provided by the parish.
4. The classes would cover the basics of the sacrament, the impediments, the responsibilities and duties.
5. If after all of this, there was a marriage, then the Church would assume validity (assuming the marriage was consumated).
Hard to argue with the Church and cry "foul" or "annulment" when all the externals match up. If one of the spouses goes "crazy" after marriage, then the only option that the "normal" spouse has is prayer and fasting - that God would sort things out.
Outside of this, I'd fear that the "normal" spouse is in a very difficult situation from which there is no solution, except death. As the jews asked Christ, after He reiterated that divorce was wrong, "If the case of a man with his wife be so, it is not expedient to marry." (Matt 19:10)
Christ responded: Who said to them: All men take not this word, but they to whom it is given. 12For there are eunuchs, who were born so from their mother's womb: and there are eunuchs, who were made so by men: and there are eunuchs, who have made themselves eunuchs for the kingdom of heaven. He that can take, let him take it. (Matt 19:11-12)