The בעל המאור in Pesachim gives a cryptic answer, he states that it is only זכר למקדש and therefore there is no שהחיינו. This is difficult because we find that we do say שהחיינו on מצות דרבנן. The Brisker Rav explains the בעל המאור as follows.
There are different types of מצות דרבנן
1. A new mitzvah like Channuka
2. A mitzva that used to be min hatorah that no longer applies min hatorah. The Chachamim tell us to do the same exact mitzva m'drabbanan, for example, מרור nowadays
3. A new mitzva זכר למקדש
The Brisker Rav claims that according to the Baal Hamaor, ספירת העומר is category 3, it is a new mitzva זכר למקדש. He brings a number of interesting proofs.
The gemara in Menachos comments that some of the Amoraim only counted days and not week, because it is only זכר למקדש. If it was in category 2 we would say that the mitzva needs to be done exactly as it was done min hatorah. However, if it is a new mitzva זכר למקדש we can say that they only instituted days and not weeks and understand the opinion of these Amoraim.
After counting we say הרחמן הוא יחזיר וכו. We don't say this after we eat maror or after we shake a Lulav (after the first day of Succos which is only מדרבנן). The reason is because we are being מקיים the same mitzva only מדרבנן. However, ספירת העומר is a new mitzva specifically instituted זכר למקדש and therefore we mention that we hope that the Avoda in the Beis Hamkidash is restored becuase that is the whle purpose of the mitzva.
If ספירת העומר is a new mitzva זכר למקדש we can also understand why we don't say שהחיינו, we are not happy that we are doing this mitzva, rather it reminds us that we don't have a Beis Hamikdash.
We should keep this idea in mind during sefira as we prepare for Kabbalas Hatorah, the lack of a Beis Hamikdash should be bother us and hopefully we will all be able to celebrate Pesach Sheni in the Beis Hamikdash.