Simply speaking, even if commandments require intention, in any case one does not need explicit intention on the Seder night, based on the words of the Chayei Adam in section 68, and this is his language: “But if one recites the Shema in the way we recite it as part of the order of prayer, and likewise if one ate matzah, or blew the shofar, or took the lulav, even though he did not intend to fulfill his obligation, he has fulfilled it, for he is doing it for that very reason, in order to fulfill his obligation, even though he is not consciously intending it.” End quote. And certainly this reasoning applies to a commandment over which one first recites a blessing. And this can also be combined with regard to some of the commandments of the night that are rabbinic.

Rabbi Asher Weiss expanded on the words of the Chayei Adam very nicely and in well-reasoned language in Minchat Asher, Pesachim, section 81, that this is not because of an unspecified act being considered for its proper intent, but because of the form of the commandment itself.

I would note that the Mishnah Berurah brings the words of the Chayei Adam and in any case writes that one should intend this from the outset. But if my memory serves me correctly, according to the reasoning of Minchat Asher, even from the outset no intention is needed.

First, I was asked about the proper religious focus for this night, and intention is certainly appropriate even if it is not indispensable.
Second, this itself is the law of an unspecified act being considered for its proper intent. The rule of “unspecified is for its proper intent” was originally stated (at the beginning of tractate Zevachim and in chapter 2 of tractate Gittin) regarding proper intent, not regarding intention generally, but many halakhic decisors apply something similar to “unspecified is for its proper intent” to intention as well.
I am not familiar with what Minchat Asher says, but what you described here is exactly the reasoning of “unspecified is for its proper intent” applied to intention.