Sukkah 25a says חולין ומשמשיהן פטורין מן הסוכה - Sick people, as well as those taking care of them, are Patur from Sukkah (See 26a that this applies even if it is not a life threatening illness, such as a bad headache). Being as young children often have the Din of a Choleh (See Orach Chaim 328:17 for example) would someone taking care of their young children be Patur from Sukkah?
There are two comparisons they need to be examined:
1. between bitul asse deorayta and shvut deshabbat (davka shvut de amira lenochri),
2. between cooking meal and taking care of an ill person. In conclusion, the first comparison is hard to decide, but the second comparison is not credible.
Shulchan Aruch Orach Chayim siman 640 sayif 3:
ויש מי שאומר שאין המשמשים פטורים אלא בשעה שהחולה צריך להם.
Some poskim says that caretakers are exempt only at the time in which the ill person needs their help (not 24/7). (Mishna Berura restricted this stringency for who is nursing a Non life threatened patient. ).
The leniency is a bitul mitsvat asse deorayta by shev veal taasse.
Thus concerning Succa a sick person in life threatening state and his caretaker have the same status of entire ptor. But a sick person in non life threatening state and his caretaker are not equivalents. The sick person is entirely patur but the caretaker is partially patur.
In 328, 17 RMA (see Bet Yosef at the end of the siman(1)) in name of Rabenu Yerucham netiv 12, Chelek 9, Ramban and Rashba stated that to cook a meal for toddlers is equivalent to care of a non life threatening illness patient and we can ask a non-Jew to make it . (See Aruch Hashulchan paragraph 20) The question is about cooking and processing food on Shabbat.
הגה: מותר לומר לעכו"ם לעשות תבשיל לקטן שאין לו מה לאכול דסתם צרכי קטן כחולה שאין בו סכנה דמי (רבינו ירוחם נתיב י"ב חלק ט' ורמב"ן ורשב"א)
For Chole Sheen Bo Sakana in Shabbat, for a part poskim, the leniency is only to seek help from a non Jewish person, but not from Israel even not by the sick himself, even Isur Derabanan when there is no danger for a member (Sakkanat evar).
`Being as young children often have the Din of a Choleh (See Orach Chaim 328:17 for example) would someone taking care of their young children be Patur from Sukkah.
Caretaker of a toddler is not similar to caretaker of an ill person , for chilul Shabbat only there is a similarly between processing food and mel'achot for refuah.
The heiter of Rabenu Yerucham is only to cook meal.
To cook a meal is not similar to taking care of an ill person. From the din of Rabenu Yerucham concerning cooking meals for children, we cannot learn that cooking meals generate a ptur from succa.
כתב ר"י בח"ט ני"ב שהרמב"ם מתיר בישולי א"י לחולה שאין בו סכנה והרשב"א עשה לו סמוכות והעלה בידו דאיסור בישולי א"י שהוא איסור קל משום שמא יבא לאכול עמו הותר בחולה שאין בו סכנה בשבת עכ"ל ומ"ש בשם הרמב"ם הוא ממה שכתב בפ"כ חולה שאין בו סכנה עושין לו כל צרכיו ע"י א"י כיצד אומרים לא"י לעשות לו והוא עושה לבשל לו ולאפו' וכו'
The quote of the Bet Yosef doesn't contain an explicit mention to a child who has no food, as it seems from the words of the RMA.
Would the Gemara say this?
Does the din by Shabbos that children and their needs being considered to be like sick people (that allows some kulos in regards to Shabbos) then turn into a patur from Sukkah?
--> no there is no link between food processing and nursing.
Where does the din of children being like sick people come from?
--> From Rabenu Yerucham, since this is not explicitly said.
I finally found an answer inside. Nitei Gavriel writes that if one is needed to take care of their child outside the Sukkkah then they are indeed Patur.
(I asked this question to Rabbi Rudinsky about two years ago (shortly after posting the question here as a curiosity) and he said I was still Chayav in Sukkah but I no longer remember all the specifics. It could be that if there was no other choice but for me (and not my wife or anyone else) to watch my young children outside of the Sukkah he would agree I'd be Patur.)