Why did Achashverosh wait until the 7th day? | Megillas Esther Explained (1:10)
From the book Megillas Esther Explained (Mosaica Press)
Source: From the commentary on Esther (1:10) in my soon to be published book entitled Megillas Esther Explained by Yisroel Dov Levene (Mosaica Press).

(י) בַּיּוֹם֙ הַשְּׁבִיעִ֔י כְּט֥וֹב לֵב־הַמֶּ֖לֶךְ בַּיָּ֑יִן אָמַ֡ר לִ֠מְהוּמָ֠ן בִּזְּתָ֨א חַרְבוֹנָ֜א בִּגְתָ֤א וַאֲבַגְתָא֙ זֵתַ֣ר וְכַרְכַּ֔ס שִׁבְעַת֙ הַסָּ֣רִיסִ֔ים הַמְשָׁ֣רְתִ֔ים אֶת־פְּנֵ֖י הַמֶּ֥לֶךְ אֲחַשְׁוֵרֽוֹשׁ׃

(10) On the seventh day, when the king was merry with wine, he ordered Mehuman, Bizzetha, Harbona, Bigtha, Abagtha, Zethar, and Carcas, the seven eunuchs in attendance on King Ahasuerus,

On the seventh day – this is the last day of the seven day feast and therefore was the climax. It also meant that when the king's heart was merry with wine – the king was more intoxicated as a result of the build-up of many days of wine coupled with his happiness seeing how everything was going according to plan. He therefore erred and acted without thinking things through properly.
This is why in halacha a Rov cannot pasken new (not simple) questions after drinking even a smallish amount of wine, the amount of a reviyis.

אָמַר רַב יְהוּדָה אָמַר שְׁמוּאֵל: שָׁתָה רְבִיעִית יַיִן — אַל יוֹרֶה.

Rav Yehuda said that Shmuel said: If one drank a quarter-log of wine, he may not issue a halakhic ruling, as the wine is liable to confuse his thinking.

The Gemora says that a person can be recognized for who they really are when they are in a state of drunkenness.

בִּשְׁלֹשָׁה דְּבָרִים אָדָם נִיכָּר: בְּכוֹסוֹ, וּבְכִיסוֹ וּבְכַעְסוֹ. וְאָמְרִי לֵיהּ אַף בְּשַׂחֲקוֹ.

Rabbi Elai said: In three matters a person’s true character is ascertained; in his cup, i.e., his behavior when he drinks; in his pocket, i.e., his conduct in his financial dealings with other people; and in his anger. And some say: A person also reveals his real nature in his laughter.

The king did not pick his ordinary servants or attendants for his mission to send to Vashti, but אָמַ֡ר לִ֠מְהוּמָ֠ן בִּזְּתָ֨א חַרְבוֹנָ֜א בִּגְתָ֤א וַאֲבַגְתָא֙ זֵתַ֣ר וְכַרְכַּ֔ס שִׁבְעַת֙ הַסָּ֣רִיסִ֔ים הַמְשָׁ֣רְתִ֔ים אֶת־פְּנֵ֖י הַמֶּ֥לֶךְ אֲחַשְׁוֵרֽוֹשׁ the seven chamberlains who ministered in the presence of King Achashverosh – he asked very high up officials, from his inner circle. Not just one of them but שִׁבְעַת֙ הַסָּ֣רִיסִ֔ים, the seven chamberlains who ministered in the presence of the King Achashverosh – all seven of them making this matter very public and high profile as he wanted to show off (which is one of the stages that happens when a person gets drunk - see my Article on Slowly getting drunk). However, this meant that refusal to heed their advice would be extremely damaging to the impression that the king was coming to project to the masses about his authority.ׁׁ
What is the significance of calling all seven of them?
We find that Paro had three advisors (Sota 10b)

אָמַר רַבִּי חִיָּיא בַּר אַבָּא אָמַר רַבִּי סִימַאי: שְׁלֹשָׁה הָיוּ בְּאוֹתָהּ עֵצָה: בִּלְעָם, וְאִיּוֹב, וְיִתְרוֹ. בִּלְעָם שֶׁיָּעַץ — נֶהֱרַג, אִיּוֹב שֶׁשָּׁתַק — נִידּוֹן בְּיִסּוּרִין. יִתְרוֹ שֶׁבָּרַח — זָכוּ מִבְּנֵי בָנָיו שֶׁיָּשְׁבוּ בְּלִשְׁכַּת הַגָּזִית

Rabbi Ḥiyya bar Abba says that Rabbi Simai says: Three noteworthy people were consulted by Pharaoh in that counsel where Pharaoh questioned what should be done with the Jewish people. They were Balaam, and Job, and Yitro. Rabbi Ḥiyya bar Abba teaches what occurred to each of them: Balaam, who advised Pharaoh to kill all sons born to the Jewish people, was punished by being killed in the war with Midian (see Numbers 31:8). Job, who was silent and neither advised nor protested, was punished by suffering, as detailed in the eponymous book in the Bible. Yitro, who ran away as a sign of protest, merited that some of his children’s children sat in the Sanhedrin in the Chamber of Hewn Stone.

– like a small Beis Din of three to cover the three possible options – one to say guilty, the other innocent, and the third to tilt to one direction, making a majority.
Why then do we find Achashverosh with seven advisors?

תַּנְיָא: כֵּיצַד אָמַר רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל? בִּשְׁלֹשָׁה מַתְחִילִין, וּבַחֲמִשָּׁה נוֹשְׂאִין וְנוֹתְנִין, וְגוֹמְרִין בְּשִׁבְעָה. אֶחָד אוֹמֵר לֵישֵׁב, וּשְׁנַיִם אוֹמְרִים שֶׁלֹּא לֵישֵׁב – בָּטֵל יָחִיד בְּמִיעוּטוֹ. שְׁנַיִם אוֹמְרִים לֵישֵׁב, וְאֶחָד אוֹמֵר שֶׁלֹּא לֵישֵׁב – מוֹסִיפִין עֲלֵיהֶם עוֹד שְׁנַיִם, וְנוֹשְׂאִין וְנוֹתְנִין בַּדָּבָר. שְׁנַיִם אוֹמְרִים צְרִיכָה, וּשְׁלֹשָׁה אוֹמְרִים אֵינָהּ צְרִיכָה – בָּטְלוּ שְׁנַיִם בְּמִיעוּטָן. שְׁלֹשָׁה אוֹמְרִים צְרִיכָה, וּשְׁנַיִם אוֹמְרִים אֵינָהּ צְרִיכָה – מוֹסִיפִין עֲלֵיהֶם עוֹד שְׁנַיִם, שֶׁאֵין הַמִּנְיָן פָּחוֹת מִשִּׁבְעָה.

The mishna teaches: Rabban Shimon ben Gamliel says that intercalation is performed in stages: The deliberations begin with three judges, they debate the matter with five judges, and they conclude the matter with seven judges. The Gemara elaborates: It is taught in a baraita (Tosefta 2:1): How does Rabban Shimon ben Gamliel say the intercalation is to be conducted? The court begins the deliberation with three judges, and they debate the matter with five, and they conclude with seven. At first, three judges convene for a preliminary discussion with regard to the necessity of adding a month to the year. If one says it is necessary to sit and deliberate the question of intercalation and two say that there is not a need to sit and continue, the single opinion in favor of intercalation is negated in its minority, i.e., it is the minority opinion, and the process ends. The baraita continues: If two say to sit and one says not to sit, the majority decision causes the process to move to the next stage, and the court then adds another two judges to them and all five debate the matter. If two say: The year needs the extra month, and three say: It does not need it, the opinions of the two are negated in their minority. If three say: The year needs it, and two say: It does not need it, the court adds to them another two judges, as the quorum for declaring an intercalation may not be fewer than seven.

We find in this Gemora the process of deciding whether it will be a leap year that starts with three, then has the discussion with five and ends off with seven deciding. Perhaps according to this, the seven were the full set of advisors, but sometimes the matter would be decided by three if easy, or five is more difficult.
Alternatively, the larger the number of such high-ranking people, was to show the importance attached to that particular mission. Hence, the possuk brings all the names of the seven that would be available should they all be needed in a most complex or important situation.
If you enjoyed this, there is the whole passuk by passuk commentary on Megillas Esther, soon to be published entitled Megillas Esther Explained (Mosaica Press).