"Do we actually know why it became tradition for men to pay?"
It was a tradition, up until I'd say the 90s.
I was listening to a R4 program this morning about equality of pay. And the consensus is that yes, women are paid less than men over time. But that's when you factor in maternity leave and women staying at home to rear their children, and taking part-time work.
There is no evidence for women being paid less than men for doing the same job (contrary to reports).
From the House of Commons Library:
Median weekly pay for female full-time employees was £558 at April 2021, based on data from the ONS Annual Survey of Hours and Earnings. This compared to £652 for male full-time employees. After adjusting for inflation, median pay for women working full-time was around 2% higher than its level in the financial crash in 2008, while median pay for men was around 8% lower. As of April 2020, the gender pay gap in median hourly pay (excluding overtime) for men and women was:
• 7.9% for full-time employees,
• -2.7% for part-time employees (meaning women tended to be paid more than men),
• 15.4% for all employees. The gender pay gap for all employees is larger than either the full-time or part-time pay gaps.
This is because a much higher share of women than men are employed part-time and part-time workers tend to earn less per hour than those working full-time.
So a difference of £652 a week, to £558 - £94 a week. Hardly a ringing endorsement for "men should pay".