[the phantom]

I have always believed that elves would not care much about gender if at all and both sexes were more or less considered equal.

Here's something that Tolkien wrote that has to do with male and female elves (HoME X, Part 3, II, Laws and Customs...)-

...the neri and nissi (that is, the men and women) of the Eldar are equal- unless it be in this (as they themselves say) that for the nissi the making of things new is for the most part shown in the forming of their children, so that invention and change is otherwise mostly brought about by the neri. There are, however, no matters which among the Eldar only a ner can think or do, or others with which only a nis is concerned. There are indeed some differences between the natural inclinations of neri and nissi, and other differences that have been established by custom (varying in place and in time, and in the several races of the Eldar). For instance, the arts of healing, and all that touches on the care of the body, are among all the Eldar most practised by the nissi; whereas it was the elven-men who bore arms at need. And the Eldar deemed that the dealing of death, even when lawful or under necessity, diminished the power of healing, and that the virtue of the nissi in this matter was due rather to their abstaining from hunting or war than to any special power that went with their womanhood. Indeed in dire straits or desperate defence, the nissi fought valiantly, and there was less difference in strength and speed between elven-men and elven-women that had not borne child than is seen among mortals. On the other hand many elven-men were great healers and skilled in the lore of living bodies, though such men abstained from hunting, and went not to war until the last need.

As for other matters, we may speak of the customs of the Noldor (of whom most is known in Middle-Earth). Among the Noldor it may be seen that the making of bread is done mostly by women; and the making of the lembas is by ancient law reserved to them. Yet the cooking and preparing of other food is generally a task and pleasure of men. The nissi are more often skilled in the tending of fields and gardens, in playing upon instruments of music, and in the spinning, weaving, fashioning, and adornment of all threads and cloths; and in matters of lore they love most the histories of the Eldar and of the houses of the Noldor; and all matters of kinship and descent are held by them in memory. But the neri are more skilled as smiths and wrights, as carvers of wood and stone, and as jewellers. It is they for the most part who compose music and make the instruments, or devise new ones; they are the chief poets and students of languages and inventors of words. Many of them delight in forestry and in the lore of the wild, seeking the friendship of all things that grow or live there in freedom. But all these things, and other matters of labour and play, or of deeper knowledge concerning being and the life of the World, may at different times be pursued by any among the Noldor, be they neri or nissi.

One thing I thought was particularly interesting was that an elf could not be both a great warrior and healer (unless during different stages of his life). Elrond fits this description quite well. Many call him a wuss because he doesn't go out on the battlefield and fight, but the text leads me to believe that he would lose his mastery of healing if he did. He could fight, he just chose not to.

but generally, I believe that Elven women were weaker than the men.

Yes, the text agrees with this but also says that the difference was less obvious than it was with mortals (until an Elf woman gave birth). But since a difference does exist, it makes sense that the men are the ones fighting most of the time. And since Elven Kings often lead their forces into battle and participate in the fight a woman couldn't really be an Elven King (or at least she wouldn't last as long as a male would, as Finwe pointed out), especially a woman who had given birth like Galadriel.

They didn't go out and command forces. They left the dirty work to their husbands.

And we're happy to do it as long as our women smile and wink at us occasionally . I guess we're kind of stupid that way (or desperate).