Per Rashi, in Noach (Genesis 7:13), בעצם היום הזה/"In the selfsame day" expresses the notion of a face-off. God's way of creating a showdown between good and evil. "I will call your bluff."
בעצם היום הזה. לִמֶּדְךָ הַכָּתוּב שֶׁהָיוּ בְנֵי דוֹרוֹ אוֹמְרִים אִלּוּ אָנוּ רוֹאִים אוֹתוֹ נִכְנָס לַתֵּבָה, אָנו שׁוֹבְרִין אוֹתָהּ וְהוֹרְגִין אוֹתוֹ, אָמַר הַקָּבָּ"ה אֲנִי מַכְנִיסוֹ לְעֵינֵי כֻלָּם וְנִרְאֶה דְבַר מִי יָקוּם:
בעצם היום הזה IN THE SELFSAME DAY — Scripture teaches you that the people of his generation said: “If we see him enter the Ark we will smash it up and kill him”. The Holy One, blessed be He, thereupon said, ‘‘I will let him enter before the eyes of everyone and we shall see whose word prevails” (Genesis Rabbah 32:8).
Noach thus isn't sneaking off - let's do this in broad daylight! What I find strange is the following. Noach should be snuck in the backdoor because he doesn't quite believe in the flood. He's not exactly the poster child for this moment of truth. Picture the scene - an evil mob gathers to kill Noach and smash the ark, and he, in full view, is checking to see if the water is really getting higher. "Is it up to my ankles yet?" What gives?
Rashi, Genesis 7:7:
מפני מי המבול. אַף נֹחַ מִקְּטַנֵּי אֲמָנָה הָיָה, מַאֲמִין וְאֵינוֹ מַאֲמִין שֶׁיָבֹא הַמַּבּוּל, וְלֹא נִכְנַס לַתֵּבָה עַד שְׁדְּחָקוּהוּ הַמַּיִם:
מפני מי המבול BECAUSE OF THE WATERS OF THE FLOOD — (מפני properly means ‘‘from before”) — Noah, also, was of those people who are wanting in faith: he believed and he did not believe that the Flood would come, and he would not enter the Ark until the waters forced him to do so (Genesis Rabbah 32:6).
The Chizkuni sums it up best (Genesis 7:7). Noach entered the ark out of desperation and NOT because God commanded him to do so:
מפני מי המבול ולא מפני הצווי.
מפני מי המבול, “on account of the waters of the deluge, and not on account of G-d’s command to enter the ark.
Did the mob notice that Noach was only a quasi-believer? If so, doesn't this spoil the powerfulness of אֲנִי מַכְנִיסוֹ לְעֵינֵי כֻלָּם וְנִרְאֶה דְבַר מִי יָקוּם?/‘‘I will let him enter before the eyes of everyone and we shall see whose word prevails” The evildoers, indeed, drown, but the last thing they perforce see is the incredulous and unwilling Noach enter the ark ...