Why do we make a bracha on tzitzis that do not have techaylas? Since without techaylas we may not be properly fulfilling the mitzvah according to the Torah, then the rule should be "sofek bracha l'kullla", i.e. skip the bracha if it is questionable?
The opening Mishnah in Perek Hatecheiles (Menachos 38a) states:
התכלת אינה מעכבת את הלבן והלבן אינו מעכב את התכלת
THE [ABSENCE OF THE] BLUE [IN THE FRINGES] DOES NOT INVALIDATE THE WHITE, NEITHER DOES THE [ABSENCE OF THE] WHITE INVALIDATE THE BLUE
(Soncino translation; capitals in original)
Rashi there explains:
ואע"ג דמצוה לתת תכלת ב' חוטין בציצית כדמפרש לקמן אפ"ה אין זה מעכב את זה ואי עביד ארבעתן תכלת או ארבעתן לבן יצא
And even though the commandment is to place two strings of techeiles on the tzitzis as is explained later on, even so they do not depend on each other. And if someone made [all] four [strings] techeiles or [all] four [strings] white he has discharged his obligation.
While this is actually subject to dispute (by tannaim and rishonim) those who make a blessing on tzitzis without techeiles are following this rule, which is codified as the law by Rambam in Hilchos Tzitzis 1:4:
והתכלת אינו מעכב את הלבן והלבן אינו מעכב את התכלת כיצד הרי שאין לו תכלת עושה לבן לבדו
The [absence of] techelet does not prevent [the mitzvah from being fulfilled with] the white strands, nor does the [absence of] the white strands prevent [the mitzvah from being fulfilled with] techelet.
What is implied? A person who does not have techelet should make [tzitzit] from white strands alone.
According to (almost) all Rishonim, the Mitzvot of Techelet and Lavan are not Meakev each other (i.e., you can do one without the other), based on the Mishnah in Masechet Menachot. There is only one (that I know of) Rishon who disagrees, the Baal Hamaor (on Rif in Masechet Shabbat). He is said to have never worn tzitzit in his life for this reason. Rav Yerucham Fishel Perla, on Rabbi Saadya Gaon's Sefer Hamitzvot, writes that if both were to be available, it would be a Bal Tigra not to wear one of them. Nevertheless, if we don't have techelet, it is not a Bal Tigra.
So, what is the status of a Tzitzit with just Lavan? The Mishna Berurah (on Hilchot Tzitzit) and Rav Soloveichik (Reshimot Shiurim) both write that it is a Mitzvah, but not Beshelemutah, it is not complete. It would be the same as saying one Shema per day, according to the Rambam. A Mitzvah not Beshelemutah is not enough to be Docheh a Lav (to override a negative commandment - here it would be Shatnez), but it is still a very big Mitzvah. It is a Mitzvah Deorayta (from the Torah) still. Regarding the question about not wearing white Tzitzit outside an Eruv (Brisk does not even rely on Eiruvin), but still making a Bracha on it, it is a very important question. I once had the opportunity to ask someone from a Brisker family and he explained they were Machmir on an Issur Sekila (a sin which has a punishment of stoning, here the desecration of shabbat), but not on Bracha Levatala (a blessing in vain). Bracha Levatala is a safek (doubt) if it is deorayta (from the Torah) or derabanan (From the Rabbis) - and definitely not an issur sekila.