Consider that many men (among others) don't consciously experience regular or frequent need for verbal or tactile emotional support as such, so they don't get a lot of practice culturing it as a skill, either as a giver or receiver, and don't often get to feel out which people in their network are going to be well-suited for it anyway.

Infrequent needs are hard to "work on" and often benefit from institutional support rather than ad hoc support: therapists, churches/etc, discussion groups, etc

It's sort of like changing tires, in that way. Now that manually changing a car tire is rare, because tires are more durable and crises can generally be remediated by calling some number on the cell phone you're certainly carrying, fewer and fewer people have actually done it and have a working, practiced familiarity with how to do it. But thankfully, roadside assistance and tow trucks are widely available and there's mostly no shame to using them now.

Supporting instutitonal access for emotional support would go along way towards helping the many people who just don't have an opportunity to "work on" building support networks the way you suggest.