The Gemorah (Moed Koton 14B) understands from the pesukim in parshas Shmini that the kohein gadol does not observe aveilus over the death of a relative. (See Sefer Ginat Egoz, page 95). "The kohein gadol all year long is compared to everyone else on yom tov" - just as on yom tov no one observes aveilus, so too the kohein gadol does not observe aveilus all year long. Rav Soloveitchik explained>[1] this comparison as follows: the Gemorah (Chagiga 5B), based on a posuk in Divrei Hayomim, comments that in the presence of Hakadosh Boruch Hu there cannot be any sadness; simcha always follows when one is in the presence of Hashem. Just as on shalosh regalim we all have an obligation to be Oleh Regel, (i.e. visit the Beis Hamikdosh and enter into the presence of Hashem), so too it is the role of the kohein gadol to be in the Beis Hamikdash all day long (Rambam Hilchos Klei Hamikdash 5,7) and supervise the offering of all of the korbonos[2]. Because the kohein gadol and everyone else on shalosh regalim are obligated to be lifnei Hashem, this engenders a mitzvas simcha, which in turn is a contradiction to observance of aveilus.
The Talmud (Yevamos 62b) tells us that many thousands of students of Rabbi Akiva died during the period of sefirah, and in the days of the Gaonim the minhag developed to observe aveilus over the great loss in Torah caused by the death of so many Torah scholars. The Zohar, however, considers sefirah to be a Biblical period of mourning unrelated to the death of these Torah scholars, and explains that it is because of this Biblical element of aveilus that we don't we recite Hallel sholeim on she'vei shel Pesach in commemoration of the neis of kriyas Yam Suf, even though we do recite Hallel sholeim to commemorate other miracles (e.g. the miracles of Chanukah). Based on this Zohar, Sefardic mekubalim introduced the minhag, followed by many chassidim today, to delay beginning the counting of sefirah on the second night of Pesach until after the completion of the seder so that the recitation of Hallel sholeim at the seder comes before we start counting sefirah. The Hallel which will be recited on the morning of the second day of Pesach after Shacharis, however, is not connected to any neis but is rather an expression of our simchas Yom Tov.[3] As such, the Sefardic mekubalim were not concerned with this recitation of Hallel sholeim during sefirah. It is not proper, however, to recite Hallel sholeim in connection with a neis during sefirah.
When an individual is tamei the halacha sometimes requires him/her to wait seven clean days before going to the mikva to become purified. When the entire Jewish nation was spiritually impure after living amongst the Egyptians for so many years, it was necessary for us to have seven clean weeks. Our counting of the Omer today is also for the purpose of purifying ourselves, as we mention in the prayer we recite after the sefirah. One who is tamei is precluded from entering the Beis Hamikdosh and therefore, by definition, is in a mild state of mourning. Simcha follows when one is in the presence of Hashem, and aveilus comes when one is unable to enter into His presence.
The establishment of Medinat Yisroel took place in the middle of the sefirah. HaGaon Rabbi Meshulem Roth wrote a Teshuva encouraging the reciting of Hallel sholeim on Yom Ha'atzmaut in commemoration of the yeshua and the nissim. Rabbi Moshe Zvi Neriah, in a letter to Rabbi Roth[4], raised an objection to this recitation based on the aforementioned Zohar and minhag which dictates that Hallel sholeim should not be recited during the sefirah period even for the great neis of Kriyas Yam Suf.
[1]See Ish Ha'Halach pages 210-211; Nefesh HaRav page 314
[2]See Nefesh HaRav page 306
[3]See Sha'ar haKollel chapter 49 section 2
[4]See Rabbi Neriah's sefer, Znif Melucha