What is the halachic reasoning for adding/deleting this line to the Kaddish? Is it unique to just Nusach Sefard and Edot HaMizrach? It seems to also be near universal amongst Dati Leumi minyanim in Israel, what is the reason for this, if there is one?
I do not think we can identify precisely what the origin of the phrase is. What we can attest to is which early sources either included it (eg Rasag, Rambam, et al) or absented it (Mahzor Vitry, Roqeah, Sefer haMinhagim, et al).
Moshe Halamish (ch. 30 of הקבלה בתפילה בהלכה ובמנהג) produces a lengthy list of sources, ranging from early siddurim, halakhic codes, commentaries on the siddur, etc. on both sides during the medieval era. He also tentatively suggests that in Italy, France, Northern Spain, i.e. lands where Christianity predominated, it was deleted due to the possible danger associated with its recitation. However he admits that this perspective is not without challenge due to the various cases of exception that can be identified.
Other than the matter of not departing from one's received nusah (which is a point that could be rallied by either side in favor of its inclusion/exclusion), I'm not sure why you assume that the different traditions on this point are rooted in a halakhic rationale. While there are many occasions on which differences in nusah are rooted in differences of halakhic opinion, I do not see this as being one of those occasions.
The Rambam when he brings down the Nusach of Tefila (at the end of Sefer Ahava), writes that one should say ויצמח פורקניה ויקרב משיחיה. However, he does not say that one should answer Amen to this.
Rabbi Shlomo Lorincz in his book, "In their Shadow" writes that Rabbi Yitzchok Zev Soloveitchik (1886-1959), better known as the Brisker Rav when attending a minyan in which the nusach was ויצמח פורקניה ויקרב משיחיה, he would not answer Amen as the Rambam writes.
As pointed out in a different answer, the Rambam says to include the phrase as part of Kaddish. It is very logical to include it as well. The whole Kaddish is praying for G-d's greatness and exaltedness to be revealed to the whole world. The time when this will be carried out will be the Messianic era. Thus, it's very appropriate to include it. The question therefore is: why do Ashkenazim not say it?
The Gemara (Sanhedrin 98b) quotes the amoraim Ulla and Rabba as saying:
ייתי ולא איחמיניה Let the Messiah come, but after my death, so that I will not see him, as I fear the suffering that will precede his coming.
(The non-bolded words are the explanation from R' Steinsaltz)
In other words, as much as we want Moshiach to come, there is also reason to hope not to actually see him in our lifetime since his coming will be accompanied with lots of suffering. Based on this, we do not add in the words "May He make the salvation sprout forth and bring His Moshiach close" to Kaddish, even though it fits with the general theme of Kaddish. As much as we are praying for salvation, this is one facet of the redemption that we would prefer not to be hastened.
It should be mentioned that although Rav Shulman used the gemara to explain the custom of ommitting the phrase, it is not clear that he was actually explaining the original source of the minhag. He may just have been giving an explanation of the custom, not the source.
(It should be mentioned that in that same gemara, Rav Yosef emphatically declares that he wants to see the coming of Moshiach and would be willing to suffer any negative consequence necessary. If Rav Shulman was in fact bringing the gemara as the original source for the custom not to say ויצמח, then presumably, those who do say ויצמח are following Rav Yosef in this matter.)