Those of you with access to a college or university library are familiar with PsycINFO, a wonderful on-line resource of the American Psychological Association that allows a computerized search by keywords of English-language studies published by psychologists, from 1872 to the present.
With Father's Day approaching, I just did separate searches for the terms "mother OR mothers OR maternal" and for the terms "father OR fathers OR paternal." I found what I expected: 97,957 studies on mothers and 35,236 studies on fathers, an almost 3 to 1 discrepancy. Similar discrepancies occur if we compare the number of Mother's Day cards and gifts sent each year to the number of Father's Day cards and gifts sent each year*. And the number of phone calls to Mom on her day versus Dad on his day is similarly out of balance**.
So, are Moms more important than Dads? Of course not. Besides what I can glean from my Internet searches, I cannot speak intelligently to the epidemiology of cards, gifts, or phone calls, but I can say something about psychology research studies. Although psychologists have long been more interested in mothers than in fathers, things have changed in recent years, with studies of fathers dramatically increasing. The focus of studies has also changed. In the past, the most typically studied topic was father absence, but many of the newer studies look at the benefits fathers afford their children. What we see is a trend consistent with the premise of positive psychology to look at what goes right in life, including in this case the benefits of fathers who are present in the lives of their children. If you look, you will find.
There are procedural challenges in studying the effects fathers have on their children. A researcher may want to tease out the separate effects of mothering and fathering, which are of course related. Some mothers may encourage some fathers to behave in certain ways, thereby creating an apparent association between fathering and child outcomes, but the real action may be in what the mothers do. (And the same caution applies when looking at the effects of mothering without taking into account what fathers do.)
Along these lines, the family's financial resources must be taken into account, because poverty - or wealth - can influence both how someone parents and child outcomes of interest, with no direct link between them.
So, per the research, what good are fathers? Carefully-done studies show that fathers have positive influences on their children throughout life, although the details differ depending on the age of the child and whether the child is a boy or a girl. And the degree of positive influence depends on whether someone is a "good" father, an obvious but important qualification. Fathers who are abusive of course do not benefit their children, and all would be better off if those sorts of fathers were not around. But most fathers are good ones, and research suggests that good fathers are those who are actively involved in the lives of their children. Active involvement is a deliberately general term, reflecting the point that there is no one way to be a good father.
Active involvement is often defined in terms of (a) engagement (directly interacting); (b)accessibility (being available); and (c) responsibility (providing resources). Actively-involved fathers have close and affectionate relationships with their children; they spend time with their children; they talk to them about things that matter; and they are the kind of person their children want to be as adults (e.g., Harris, Firstenburg, Jr., & Marmer, 1998).
In general, actively-involved fathers provide their daughters and sons with a life-long example of what it means to be a good man, a good husband, a good parent, and a good person, and their children make wiser choices in their own lives as a result of the lessons learned while growing up.
The important bottom-line from research is that actively-involved fathers have children who fare better, physically, psychologically, and socially.
A difficult issue is the one captured by the phrase quality time. The point of quality time is a good one - what one does with one's child is more important than one's mere physical presence, even if prolonged. But there are boundary conditions, and time is a quantity as well as a quality. One minute a week of active involvement with a child likely has no beneficial effects. That said, the exact parameters of quality time are not known.
Many fathers are actively involved with their children as the breadwinners for the family, meaning that the amount of direct time they spend with their children is limited. In the 1950s and 1960s, my own father fit this model of a good father. He was not often around to change diapers, to wipe away tears, or to attend PTA meetings. Instead, he was at work 10-12 hours a day, commuting 60+ minutes each way from our north suburban home to the south side of Chicago and back.  But he put the bread on the table, and he paid the tuition to the colleges my brothers and I attended.

When he was around, he also provided some of the wonderful and still memorable occasions of my childhood: Easter Egg hunts featuring clues he wrote about where the eggs were hidden in our house, frequent visits to bookstores before there was a Border's on every corner, and countless shows and concerts ranging from classical ballet to raucous rock-and-roll.
Maybe as youngsters my brother and I did not fully understand what it meant to have a father who was primarily a breadwinner, but we do now. Because of our father, we are well-educated and have reaped the benefits. Because of our father, we work hard, and we treat others well. Because of him, our lives have been much more worth living.
Looking back, I see that my father was always present, psychologically if not literally, and he was always involved. Always. Thanks Dad. I love you.
*Per Wikipedia, Mother's Day is a wildly successful commercial occasion in the US. It is reportedly the most popular day of the year to dine out in a restaurant. Every Mother's Day sees Americans spend $2.6 billion on flowers, $1.5 billion on gifts (including 8% of all jewelry purchases), and $68 million on greeting cards. Comparable statistics for Father's Day are harder to obtain, which may reflect the point I am trying to make, but they seem to be lower in each and every case.
**Indeed, the highest volume of phone calls made every year in the US is on Mother's Day. Interestingly - really! - the largest volume of collect calls made every year in the US is on Father's Day.
Reference
Harris, K. M., Firstenburg, Jr., F. F., & Marmer, J. K. (1998). Paternal involvement with adolescents in intact families: The influence of fathers over the life course. Demography, 35, 201-216.
https://www.psychologytoday.com/blog/the-good-life/201106/what-good-are-fathers?collection=66000