I'm not disputing that society was patriarchal at the time marriage was conceived, but the idea of marriage between one man and one woman isn’t "entirely patriarchal." It also follows plenty of logical, common‑sense lines.
The basic premise was that one man and one woman join together to found a family — which of course, back in the day, could only be done between men and women.
The commitment to one partner, to the exclusion of all others, was to protect the interests of that family unit - so yes ensuring that children were unquestionably born to them and only them, as you say, but also making sure that resources were directed into that single household rather than scattered across multiple partners or sets of offspring - ensuring survival and prosperity, etc.
It also had practical implications for health in that it avoids incest and inbreeding - when people know who is related to whom, there is less risk of accidentally marrying/reproducing with a sibling or other family member.
There's also evidence that two-person relationships (i.e. dyadic attachments) make for happier, more secure partnerships, because of the emotional feedback loop. They have less stress and handle conflict better. This is obviously better for public mental health, the stability of the workforce, etc.
There are good reasons why two-person marriage makes sense, and they're not all patriarchal.