No, you do not have it right. To say that at sexual relationship is necessary for marriage is not the same as saying it is entirely sufficient all by itself.
Let’s picture this couple you’re talking about…they’re living as brother and sister. A brother and sister can live like that; they cannot marry. They can love each other, care for each other, understand each other like no one else does, and yet they cannot marry. Why? Because marriage, on its more fundamental level, is a sexual relationship. Once the marriage is consummated, a couple has a sexual relationship, even if they never have sex again. And why is it different when that brother and sister smooch and hold hands than if they had a sexual relationship? You know and I know. Your wife is not your sister, friend. You know it and I know it. And what about having a sexual relationship that you are never given the right to consummate, that is never genital in nature? No frustration there, is there? Well, if you give that up by choice, that is one thing, but the Church is saying that your spouse must at least have the capacity to grant you your marital right. You always have that right. If you are in a Josephite marriage, then if you decide you want a marriage that has sex, you and your spouse have to decide to have an average sort of marriage, or else what you had may be dissolved.
But let’s turn tables…let us say that marriage does not involve a right to have sex. You get married, and your wife will not have sex with you. Is that a marriage? No, if you marry someone, they promise to have a marital relationship with you, and then you never have sex even once the entire time you are married, the promise to you has not been honored. Your spouse implied you had a right and had no way to give you your right or has decided to withhold that which the fundamental nature of marriage implies. Your wife may refuse to clean your trout, she may refuse to be nice to your mother, she may refuse to watch football with you, but she may not refuse to have sex with you entirely. She would be refusing to act married.
Honest to goodness, people are acting as if it is petty for one spouse to insist on a right to consummate their marriage. Their marriage!!! As if marriage is about “caring”, not about sex. I have news: The whole Christian life is about caring. Marriage is something else again. Marriage is about the authentic expression of the sexual faculty, something that moderns who do not like the age-old concept of marriage have totally divorced from each other in favor of either a) a nebulous sense of hearts and flowers and “I’ll be there for you” that does not belong exclusively to marriage or b) a relationship in which some aspects of the sexual faculty have been split away from intercourse open to procreation.