As for ピーピー, it is an onomatopoeia of the cry of a baby bird. So がばいばあちゃん wanted to express her grandson to be unripe and noisy just like a baby bird.

As for で済んだ, yes, it is generally used in the positive sense of "get by without ~ing".
In this context, the state of the baby's ピーピー crying is expressed far better situation than that of something other, so the cry was expressed in the positive sense like to get by without making a further more trouble or annoyance.

As for ピーピー, it is an onomatopoeia of the cry of a baby bird. So がばいばあちゃん wanted to express her grandson to be unripe and noisy just like a baby bird.

As for で済んだ, yes, it is generally used in the positive sense of "get by without ~ing".
In this context, the state of the baby's ピーピー crying is expressed as a far better situation than that of something else, so the cry was expressed in the positive sense like to get by without making a further more trouble or annoyance.

As for ピーピー, it is an onomatopoeia of the cry of a baby bird. So がばいばあちゃん wanted to express her grandson to be unripe and noisy just like a baby bird.

As for ピーピー, it is an onomatopoeia of the cry of a baby bird. So がばいばあちゃん wanted to express her grandson to be unripe and noisy just like a baby bird.

As for で済んだ, yes, it is generally used in the positive sense of "get by without ~ing".
In this context, the state of the baby's ピーピー crying is expressed far better situation than that of something other, so the cry was expressed in the positive sense like to get by without making a further more trouble or annoyance.