Is there any reason to say that its problematic to hold a child during bentching and so on... because one is thinking about not dropping what he is holding,like by Shmoneh Esrei (96:1) one should not hold anything that ones mind will be concerned about?
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The Rivevos Ephraim 6:410:1 brings the psak of Rav Eliyashiv that one may hold a child during bentching.
In Chelek 8:572:1 he was asked to explain the psak of Rav Elyashiv how its ok since there are achronim who hold by pisukei dizimrah one cannot hold anything so certainly by a doraisa one would have to avoid such a thing. Rav Ephraim Greenblatt(Rivevos Ephraim) answered he is talking about a case where the child is crying,if the child would cry and would not be able to concentrate then one may hold the child.
Sefer Avnei Yashfei chelek 5:52 wonders about women who hear kiddush while holding a child if its mutar or not. He considers one who hears kiddush is like the one who is making it(Tosfos in Brachos 20b) he seems to hold that its assur to hold a child during kiddush. He brings his Rebbi Rav Eliyashiv who holds that the gemara(which does not allow holding things) is only in regards to Shmoneh Esrei and if one is machmir by other things they are doing it only as a chumra and one is not obligated to be machmir.
The Avnei Yashfei said he was not privileged in understanding how one can be maikel against the Pri Megadim who holds the halacha is by pisukei dizimra and bircas krias shema. He ends off by saying that one must say that his intention is that it is in a makom tzarich that the mother has to hold the child to calm the child during kiddush.
(it is interesting to note,both Tshuvos use the same logic)
I recently read an interview with a daughter of Rav Ovadia Yosef ZT"L and during the interview she mentioned that her father Rav Ovadia always held a grandchild on his lap during bentching.
O Ch 191 (3) MB  says that one may not do a melacha while saying birkas hamozon. The MB says that even a תשמיש קל (a light activity) is prohibited. If holding a child is a light activity (or more) then it would be forbidden.
In O Ch 183 (12) MB , it says that one may not make a brocho while doing work and the MB and Shaar Hatziyun say that this applies to all brochos even those that are rabbinic.