Minato was a protective man. Unsurprisingly, he was a very, very protective father.
Even less of a surprise: Kushina felt smothered after Minato's initial excitement. It was a cause of great contention with the previously harmonious couple.
Minato constantly attempted to prevent even the slightest risk to Kushina and their unborn child. When Minato was away on a mission, Minato acquired the aid of his multiple friends to "supervise" Kushina (Kakashi was often left to face her wrath, as he had lacked the social skill and tact required to appease the red habanero). And when nobody was available, Minato went to the extent of hiring konoha chunin to watch and help his pregnant wife.
Minato argued that he had many enemies, especially due to the war and his significant contributions to the war effort. Also, despite always managing to remain calm and collected under enemy fire or an ambush, he didn't do well concerning the people he loved outside of a shinobi scenario.
Kushina eventually relented when she had realised that the reason Minato was so overprotective was because he had lost his parents during the last war, leaving him an orphan. He didn't want to loose his family again. Especially with her pregnancy being so difficult. Kushina didn't know whether the complication was because she was a jinchuuriki, or because she was a ninja, or whether it was because this was her first pregnancy. They almost lost their child during her first trimester, when a complication occurred, and it was only with the help of Tsunade (who was thankfully on a mission nearby) that the baby's life had been save. Tsunade was named godmother by a crying Minato when he'd found out what had almost happened.

He vowed to protect them with his life.
So, in a bid to ensure that he was there for his daughter's birth, Minato volunteered himself for every mission available (of course, with the ready approval of his team) in order to accommodate the week off he wanted so he would be available near Kushina's due date.

So when the mission came in for the destruction of the Kannabi Bridge, team Minato was otherwise unavailable. Kakashi (who had been promoted earlier than expected due to their suddenly influx of completed missions) had lead his first mission a month previous to great success, despite an initial complication regarding a disagreement between Kakashi and Obito that had eventually been resolved by Rin. Therefore, the team (despite having an available leader) opted to stay in Konoha for the birth of their sensei's much anticipated child. Besides, they didn't really want to go on such an important mission without their sensei after the disaster of Kakashi's first jounin mission.

"Minato, you need to relax" head peaking into the kitchen, Kushina could only sigh when her husband had once again situated himself on their kitchen table, configuring seals in anticipation for the birth of their daughter. He had spent the past 3 days going over his seal designs, checking for any errors despite Kushina's and Jiraiya's reassurances. Since going on leave, Minato triple checked all preparations once a day, ensuring Tsunade was on standby, Jiraiya was still in the village (who had also requested to stay in the village for Kushina's delivery in case the seal experienced any problems Minato was unable to rectify) and that birthing location was still secure. The 3rd hokage allowed such demanding circumstances because he really want a Kyuubi attack on top of the 3rd great shinobi war. Also, 2 sannins make for a very compelling argument to the man who had practically raised them. So Minato was ready for anything.

Anything but the sound of his wife gasping from the kitchen. Running (as he had forgotten about the Hiraishin in his panic) and almost tripping over his feet, Minato faced the sight of his wife standing in a puddle of liquid, holding her stomach with both hands. Husband and wife looked at each other with similar expressions of equal amounts excitement and fear before Minato came to his senses and ran for the birthing bag. He hastily created multiple shadow clones to relay the news to all the parties involved, before he was rushing his wife out of the house. Kushina had to laugh despite her worries.
"Minato, you need to calm down. It's like you're the one about to have your body ripped open by our child"
Minato only glared, and Kushina allowed herself to be ushered out of the house.

The birth was as expected. Expected by Minato, anyway. It was a wreck. A disaster, and not just because Kushina had actually broken his hand when the first bad contraction hit.
It all went to hell when Kushina started pushing. The seal had started to become unstable, and it took the combined efforts of Minato and Jiraiya to stabilise it for the birth. Tsunade (who had thankfully been Kushina's primary physician since the discovery of the pregnancy) also had to spend a good few hours healing Kushina's uterus in between contractions.
It was a difficult 23 hours of labour, especially for Kushina, who had to endure the Kyuubi chakra travelling through her whenever the seal weakened.
But in the end, it was all worth it.
Their child. The light of their life. She was their 'Hikari'.

When Minato held his beautiful little girl in his arms, he promised himself that he he would do whatever it took to protect his family and any other children he and Kushina would have.
He needed to prepare for the worst. It was only through luck that Tsunade had been there when they had almost lost their baby. It was through luck that Jiraiya had been available for the birth, providing the extra help he had needed to stabilise the seal like that. If anything else had gone differently, he would've lost his child before he had even found out. God forbid, the Kyuubi could have escaped. Minato couldn't risk that. He needed to start planning. For anything.
He now knows what could happen. What he could do.

In another life, he had returned home from a normal mission to a crying wife, sobbing and apologising to the lost child they had never gotten to hold. In a another life, this extra bit of knowledge would have saved a little boy from living a miserable, loveless childhood.
Maybe in this life, it will.