The turning point for what you're referring to - the Tender Years doctrine - is actually decades before women gained the right to vote!

The common-law English legal standard prior to the nineteenth century was for fathers to receive custody, no matter what. Parliamentary divorce was a costly process only available to the wealthy, which required the details of the case to be presented to and dissected by members of the government; it would at least result in both ex-spouses being able to marry again, where a legal separation through the religious Doctor's Commons did not allow remarriage. It was almost always initiated by the husband: women needed to prove a pattern of extreme abuse on their husband's part as well as flagrant infidelity or desertion, while a man who could prove without a doubt that his wife had been unfaithful to him sexually would be much more likely to have his responsibility for her ended. (The southern American colonies followed this standard, while in the non-Anglican north divorce was seen as preferable to the social disruption of marital violence or sinful adultery, and so somewhat more accepted and easier to get for both women and men.)

Following a divorce, the former wife was in a bad situation. She had none of the property she'd brought to the marriage, and was no longer being supported by her husband, so she had to hope that either her birth family or the man she had been involved with would take her on. She had also been publicly branded as disreputable - a whore, essentially - and would be shunned, which would give both her family and her lover pause, since associating with her could tarnish their own reputations. At least in the case of a parliamentary divorce, she had the hope of remarrying, which could mend her problems somewhat; if the case had gone through the ecclesiastical court, she was simply ruined forever.

At this time, children were viewed in the legal system as the property of their father, and it was also considered in their best interest to remain in paternal custody (except in cases where the fathers were exceptionally abusive and neglectful, at least in America). Even in a normal marriage, mothers weren't automatically considered their children's guardians after the father's death unless it was stipulated in his will, and fathers could apprentice their children wherever they wanted without the mother's consent. Due to both the the lack of amicability in divorces and the social stigma placed on the heads of women who engaged in extramarital affairs against their husbands' wishes, the fathers were typically unlikely to allow ex-wives any contact with their children.

In the late eighteenth century, two things were happening that affected this. One is that divorces were becoming more common than ever. Though they were still difficult and embarrassing to get and still carried a social stigma, the number of divorce cases rose and rose. (As did court cases charging "criminal conversation" - husbands filing suit against men they believed to be sleeping with their wives.) In Massachusetts, for instance, only six divorce petitions were filed between 1705 and 1714, but there were 46 between 1765 and 1774, and almost 90 between 1775 and 1784! In Britain, there were a number of high-profile divorces between aristocratic spouses that would have also helped to raise awareness of the possibility.

The other is that the views of motherhood and childhood were shifting. Now, don't jump to the old "people saw children as tiny adults and didn't care about them that much" workhorse - they did love their babies, nurture them, and look at them as more than just economic assets to the household, and mothers were seen as playing an important part in raising their children. But in the period following the American Revolution, the status of this role shot up. Sending your infant out of the home to be raised for months or years by some woman in the country was out; hiring a live-in wet nurse or, better yet, nursing your child yourself was in. Motherhood was idealized and even sanctified. Instead of being stern and disciplinary, women were to treat their children sweetly, encouraging them to be virtuous and molding them gently into respectable adults, and if you were a genuine Good Mother, the kids would turn out perfectly.

Add both of these things together, and you get a stage set for someone to argue that women who divorce are not the worst thing ever and that it's better for their children to remain in their custody, which is what happened with Caroline Norton in the 1830s.

Caroline Norton was one of the most “visible” women of her day. She was a cynosure of public attention for nearly half a century following her formal introduction to fashionable society during the London season of 1826. Recognition and influence might never have been more than the result of physical beauty nor have extended beyond Almack’s and its immediate environs had she not also been a prolific author of the sentimental verse and melodramatic tales popular with fashionable readers. Her initial literary impact, in turn, might never have been felt beyond the world of ton and its aspirants had she not, in 1831, attracted the attention of a man soon to become prime minister. Her relationship with Lord Melbourne led to a sensational scandal after which she would never be in the position to benefit from the wisdom of her own aperçu: “Obscurity is a thicker shield than virtue.” Even though her husband’s charge of adultery was not proven, Norton was left with a maculate reputation in a marital no-woman’s-land. In a relatively short period and well before the age of 30, her public persona had changed dramatically: from one of the beautiful Sheridan sisters, whose literary precocity established her as a worthy successor to her grandfather, Richard Brinsley Sheridan, to a scandalous woman separated from her husband and excluded from the drawing rooms of polite society.

(from The Narratives of Caroline Norton, Randall Craig)

Rather than go for a divorce, Norton's husband chose to sue Lord Melbourne for criminal conversation (term explained above). A crim con case was a huge embarrassment for everyone involved, and sent the message that a husband was more interested in punishing his wife and her lover (or purported lover) than in getting rid of a spouse he could no longer stand, and this one was no exception. George Norton was physically abusive, and he and Norton had separated and come back together several times, but after the crim con case was found in Melbourne's favor neither was allowed to sue the other for divorce. George kept her children from her, and because they were still legally married, she was compelled to give him all of her property as well as the money she made from her writing - not just what she had on her, but what she made in the future.

Unlike many other women who'd been badly done by in unofficial or official separation proceedings, Norton was a popular and talented writer with influential friends, and published pamphlets expressively titled The Natural Claim of a Mother to the Custody of her Child as affected by the Common Law Right of the Father and A Plain Letter to the Lord Chancellor on the Infant Custody Bill. She was supported by the progressive MP Sir Thomas Talfourd, who introduced bills to Parliament based on the points she laid down - the second of which passed as the Custody of Infants Act 1839, not forcing any custody decisions but just giving the courts the power to grant custody of children under seven and/or access to older ones to the mother, a huge blow to the conventions that forced women to choose between their own safety and the ability to have contact with their children. A second blow came in 1873 with another Custody of Infants Act raising the age to sixteen.

From the early nineteenth century, American judges were more likely than English ones to place children with the mother, citing the best interests of the child in their decisions when the father was abusive or the mother did not seem to be immoral, working on the idea of the same glorification of motherhood described several paragraphs ago. Divorce was more available to people lower down the social ladder than in Britain, and continued to become more so; working- and middle-class fathers were less likely to be able to hire appropriate staff to care for children of "tender years", which helped to influence more judges to rule that young children required their biological mothers. To combat unofficial separation agreements that could result in children being left in legal limbo, individual states began to pass laws allowing women to appeal to courts to determine custody, with a presumption of equality between father and mother (rather than absolute paternal rights) and a tendency to assume that women's natural instincts for motherhood made their custody in children's best interests; by the middle of the century, it was fairly standard for "tender years doctrine" to be followed in cases of custody challenged in the courts, especially for female children, though on the flipside it was often assumed that it was in the best interest of older sons to be raised by their fathers, since they needed a suitably masculine environment.

I hope I've fully illustrated the progression here! Far from having always had an edge in custody cases, women were originally very much disadvantaged in this area.

Some further reading:

Randall Craig, The Narratives of Caroline Norton (2009)

Michael Grossberg, Governing the Hearth: Law and the Family in Nineteenth-Century America (1985)

Wendy Moore, Wedlock: The True Story of the Disastrous Marriage and Remarkable Divorce of Mary Eleanor Bowes, Countess of Strathmore (2009)

Joan Perkin, Women and Marriage in Nineteenth-Century England (1988)