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Imagine a Masjid where the Imam pauses during Salah and the entire congregation waits so a toddler can finish his game. Imagine a Masjid where an Imam leads Salah while he holds a child in his arms. Imagine a Masjid where the cry of a baby changes the Imam’s intention and shortens the prayer for the entire congregation.

This was the Masjid of the Prophet Muhammad, peace and blessings be upon him.
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The Messenger of Allah came out to us for one of the two later prayers [Thuhr Asr], carrying Hasan or Hussein. The Prophet then came to the front and put him down, said Takbir for the prayer and commenced praying. During the prayer, he performed a very long prostration, so I raised my head and there was the child, on the back of the Messenger of Allah, who was in prostration. I then returned to my prostration. When the Messenger of Allah had offered the prayer, the people said: “O Messenger of Allah! In the middle of your prayer, you performed prostration and lengthened it so much that we thought either something had happened or that you were receiving revelation!” He said: “Neither was the case. Actually, my grandson made me his mount, and I did not want to hurry him until he had satisfied his wish" (Reported by Nasaa'i, Ibn Asaakir, and Haakim).

The Messenger of Allah would pray holding Umaamah bint Zaynab bint Rasoolillaah.   He would put her down when he prostrated, and then pick her up again when he stood up (Reported by Bukhari and Muslim)

The Prophet said: “When I stand for prayer, I intend to prolong it, but on hearing the cries of a child, I cut it short, as I dislike to trouble the child's mother" (Reported by Bukhari).

These scenarios illustrate how natural it was for babies, toddlers, and children to be part of the Prophet’s Masjid. They show us that the Prophet understood the nature of children and cared for their needs and comfort in the mosque.

Do we witness these scenarios in our Masjids today? We see a child grabbed and told to “sit down and be quiet!” because he was running between the rows. Or we hear “sister, can you please go outside” because her baby is crying. Or we read signs that say “No children in Masjid area”.
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Children running in the hallway or playing basketball in the parking lot are not signs of a rowdy Masjid or misbehaving children.
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This is the nature that Allah created in children - they overflow with energy and enthusiasm
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Here is an example of how the Prophet dealt with Anas bin Malik, who served him as a boy:

    The Prophet sent Anas on an errand and Anas reported that “I went until I came upon children playing in the street. Then the Messenger of Allah arrived and he caught me by the back of my neck from behind. As I looked at him, I found him smiling, and he said, “Unays (Anas’s nickname), did you go where I ordered you to go?” I said, “O Messenger of Allah, yes, I am going.” Anas said further, “I served him for nine years, but I do not know that he ever said to me about anything I did, ‘why I did that, or about anything I had neglected, why I had not done that” (Reported by Muslim).

The Prophet did not scold Anas for forgetting his errand or for anything else in nine years.
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Allah does not punish children for their misbehavior but Allah does hold us accountable for how we treat children.