The gemara[2] describes the morning berachot as being recited as one performs the particular action to which each beracha relates (e.g., opening one’s eyes, putting on shoes, washing hands, putting on tzitzit, etc.). Nowadays, however, we make the Birkot HaShachar, which praise HaShem for providing us with our physical needs, at one time, usually at the beginning of davening. The Shulchan Aruch[3] provides two reasons for not reciting these berachot sooner: 1) Our hands are considered dirty when we get up, making it improper to make a beracha; 2) since some people cannot make the berachot themselves, they can listen to the chazan say them in shul. Your query revolves around the question of whether the beracha on netilat yadayim should remain at its natural time, when washing, or can also be moved along with the series of berachot recited in shul.
The Beit Yosef[4] cites two approaches to the matter. The simpler one, which he accepts,[5] is that the beracha should not be delayed so much, as you argued. Birkot hashevach,[6] such as the Birkot HaShachar, are made after exposure to or experiencing the matter that one is praising, whereas birkot hamitzva, such as netilat yadayim, are made before performing the mitzva.
Interestingly, even according to this approach, the beracha on netilat yadayim is unlike other birkot hamitzva. It is recited after washing, and not before, because before washing, one’s hands are likely soiled in a manner that precludes making berachot.[7] However, there is still logic to make the beracha either before one wipes his hands dry or soon thereafter, as opposed to after a long break until davening.[8]
The Beit Yosef justifies the minhag (which was more prevalent among Ashkenazim) to make the beracha on netilat yadayim in shul, by citing the Rashba’s reason for netilat yadayim as support. The Rashba[9] says that as one embarks on the day as a new creation, he must thank HaShem and wash his hands like a kohen, in preparation for expressing thanks in the Birkot HaShachar. Since the washing is related to the Birkot HaShachar, just as the latter are recited in shul, the beracha on netilat yadayim is also made there. Based on this, the Perisha[10] concludes that this beracha is not a standard birkat hamitzva.
The Rosh[11] says that netilat yadayim and its beracha were instituted as a preparation for tefilla in addition to being a prerequisite for a bread-based meal. According to this opinion, which the Chayei Adam[12] says we should not dismiss, if one washed his hands upon waking and later went to the bathroom before davening, it would be the handwashing after using the bathroom that would require the beracha. The Chayei Adam notes that this is a problem only if one wants to make a beracha on netilat yadayim upon waking when he expects to use the facilities between netilat yadayim and davening.
The Gra[13] goes further, arguing that due to the connection between netilat yadayim and tefilla, the appropriate time for the beracha is specifically before tefilla. (Admittedly, he appears not to require another netilat yadayim if he does not use the facilities in between). However, the Gra’s opinion is extreme in this matter. This is evidenced by the fact that he accepted the Rosh’s view to such an extent that he requires netilat yadayim with a beracha before Mincha and Ma’ariv as well,[14] a practice that we do not observe.
In these matters, we would suggest that Ashkenazim (before davening) and Sephardim (after washing)[15] follow their respective classical minhagim.[16] The question that arises for Ashkenazim is what to do when there is a longer than usual wait between netilat yadayim and tefilla. The Chayei Adam[17] suggests that the person go to the bathroom again, making the beracha after that. However, the Bi’ur Halacha[18] raises the issue that according to the aforementioned Rashba, the beracha will not relate to the netilat yadayim that requires a beracha, which is the one upon awakening. This appears to be a concern if a long time elapses, even when one did not use the facilities in between.[19] The Rama[20] leans toward making the beracha earlier in this case, whereas the Bi’ur Halacha appears to side with the Chayei Adam. The safest thing, in the event of an expected long break, is to make the beracha on netilat yadayim at home, followed by Birkot HaShachar, including Birkat HaTorah, which marks the beginning of davening.[21]