The Shulchan Aruch in O.C. 677:3 writes:
יש אומרים שאע"ג שמדליקין עליו בתוך ביתו, אם הוא במקום שאין בו ישראל מדליק בברכות
Some say that even though they are lighting for a person at his home, if he is in a place where there are no Jews he should light with the blessings.
The Rema adds
And this is our practice.
This is a person who is ...
The Tur (OC 672) says that lighting during the day doesn't work because (alluding to Chulin 60b) שרגא בטיהרא מאי אהני – "a candle in the daytime, what does it accomplish?" In other words, flames aren't noticeable in broad daylight that the miracle would be publicized.
The Rivevos Ephraim 5:428 addresses this . He brings Mishna Berurah 671:30 which notes that if one lights in a place that is less than 20 amos and then moves it to another place which is also lower than 20 amos one is yotzeh,since both places is kosher. He also brings the Shaar Hatzion 34 brings the Machtzis Hashekel and says this is pashut(obvious). However ...
R. Moshe Feinstein (Igros Moshe Orach Chaim 4:105:7) rules pretty definitively that one should light with a bracha, even if there are no other people present to allow pirsumei nisa.
He is not particularly concerned with the Magen Avraham's chumra of waking up other members of the household in order to light late at night.
Your hypothesis is wrong.
a great miracle happened on Purim, too. Why don't we publicize this miracle?
We do, by reading the m'gila. See e.g. https://judaism.stackexchange.com/a/64482 and Different miracle-publicization strategies on Chanuka and Purim.
there's no mitzvah on Pesach to "publicize" it.
There is: we drink four cups of wine at the seder. ...
I would say that since this commemorates the miracle of lighting the menorah in the Temple (as explained in many locations) we have to light in the same way that the menorah was lit in the temple. This was done at night and not during the day.
Additionally, the pirsumei nisa is mainly at night when the candles can be seen in the dark. They would not really ...
See the Beit Yosef on the Tur Orach Chaim 671 (corresponding to Shulchan Aruch 671:7).
"כיון שעתה שיד האומות תקפה עלינו ואין אנו יכולים לקיים המצוה
כתיקונה ומדליק כל אחד בפתח ביתו מבפנים, ואין כאן
פרסומי ניסא כי אם לבני ביתו בלבד, לזה הנהיגו להדליק בבית הכנסת לקיים פרסומי ניסא".
We at least see from here that the Beit Yosef holds that ...
From Rabbi Bogomilsky, posted on Chabad.org:
"נר חנוכה מצוה להניחה על פתח ביתו מבחוץ"
“The requirement is to place the Chanukah light by the doorway of one’s house, from the outside.” (Shabbat 21b)
QUESTION: Rashi writes that this is because of pirsumei nisa — to publicize the miracle. Why on Chanukah is pirsumei nisa emphasized as a prerequisite for ...
Interesting question. I have no source, but a possibility may be that the other miracles you mentioned (besides the correct response that we have a different way of expressing it by reading the Megillah) is as follows:
With the neis of Purim, the entire kingdom was aware of the impending crisis and the "turnaround," so the neis of Purim was well known (...