When interpreting a pasuk, it is critical to understand the context.
One important piece of this context is that there were local courts established in the gates of each town. Thus, in Devarim 16:12:
שֹׁפְטִים וְשֹׁטְרִים תִּתֶּן לְךָ בְּכָל שְׁעָרֶיךָ אֲשֶׁר ה אֱלֹהֶיךָ נֹתֵן לְךָ לִשְׁבָטֶיךָ וְשָׁפְטוּ אֶת הָעָם מִשְׁפַּט צֶדֶק:
You shall set up judges and law enforcement officials for yourself in all your cities that the Lord, your God, is giving you, for your tribes, and they shall judge the people [with] righteous judgment.
These are local courts, because they are set up בְּכָל שְׁעָרֶיךָ, in all your gates ["cities"].
With that as background, when we encounter Devarim 17:8:
כִּי יִפָּלֵא מִמְּךָ דָבָר לַמִּשְׁפָּט בֵּין דָּם | לְדָם בֵּין דִּין לְדִין וּבֵין נֶגַע לָנֶגַע דִּבְרֵי רִיבֹת בִּשְׁעָרֶיךָ וְקַמְתָּ וְעָלִיתָ אֶל הַמָּקוֹם אֲשֶׁר יִבְחַר יְהֹוָה אֱלֹהֶיךָ בּוֹ:
If a matter eludes you in judgment, between blood and blood, between judgment and judgment, or between lesion and lesion, words of dispute in your cities, then you shall rise and go up to the place the Lord, your God, chooses.
we understand that the pasuk is not directed towards the litigants, but towards the local judges. That is, there are דִּבְרֵי רִיבֹת בִּשְׁעָרֶיךָ, words of dispute in your gates ["cities"], such that the local judges cannot come to a conclusion. Thus, כִּי יִפָּלֵא מִמְּךָ דָבָר, "a matter eludes you" [the judges] in judgement.
In such a situation, the case goes to the Supreme Court, which is a Sanhedrin, or a kohen Gadol, or whatever the Supreme Court is in those days.
This parallels the court system set up by Moshe in parashat Yitro (Shemot 18), in which there were lesser judges who judged the easy minor cases, while the difficult cases would be taken to Moshe.
When interpreting the subsequent pesukim, Chazal kept the same actor throughout. Since the ones who were unable to ascertain the laws were the judges, it is the judges rather than the litigants who go to inquire in pasuk 9:
וּבָאתָ אֶל הַכֹּהֲנִים הַלְוִיִּם וְאֶל הַשֹּׁפֵט אֲשֶׁר יִהְיֶה בַּיָּמִים הָהֵם וְדָרַשְׁתָּ וְהִגִּידוּ לְךָ אֵת דְּבַר הַמִּשְׁפָּט:
And you shall come to the Levitic kohanim and to the judge who will be in those days, and you shall inquire, and they will tell you the words of judgment.
And once the judges are informed what the words of judgement should be, that is what they should render as their own words of judgement.
Now, in pasuk 10 and 11, reference is made to action (וְעָשִׂיתָ). This means how you [the judge] should render judgement in terms of what the practical action should be, rather than meaning how the litigant should act.
וְעָשִׂיתָ עַל פִּי הַדָּבָר אֲשֶׁר יַגִּידוּ לְךָ מִן הַמָּקוֹם הַהוּא אֲשֶׁר יִבְחַר יְהֹוָה וְשָׁמַרְתָּ לַעֲשׂוֹת כְּכֹל אֲשֶׁר יוֹרוּךָ:
And you shall do according to the word they tell you, from the place the Lord will choose, and you shall observe to do according to all they instruct you.
עַל פִּי הַתּוֹרָה אֲשֶׁר יוֹרוּךָ וְעַל הַמִּשְׁפָּט אֲשֶׁר יֹאמְרוּ לְךָ תַּעֲשֶׂה לֹא תָסוּר מִן הַדָּבָר אֲשֶׁר יַגִּידוּ לְךָ יָמִין וּשְׂמֹאל:
According to the law they instruct you and according to the judgment they say to you, you shall do; you shall not divert from the word they tell you, either right or left.
The importance of this is in establishing a consistent rule of law from a central authority. Even if you, the judge, disagrees, you may not divert.
Finally, what if one refuses to listen? Again, this is taken to refer to the local judge. If, despite the ruling of the Sanhedrin, he persists in his insistence that he is right, and continues to rule practically (אֲשֶׁר יַעֲשֶׂה) contrary to the Sanhedrin, then he is executed, as a way of preserving the rule of law. Thus, pasuk 12:
וְהָאִישׁ אֲשֶׁר יַעֲשֶׂה בְזָדוֹן לְבִלְתִּי שְׁמֹעַ אֶל הַכֹּהֵן הָעֹמֵד לְשָׁרֶת שָׁם אֶת ה אֱלֹהֶיךָ אוֹ אֶל הַשֹּׁפֵט וּמֵת הָאִישׁ הַהוּא וּבִעַרְתָּ הָרָע מִיִּשְׂרָאֵל:
And the man who acts intentionally, not obeying the kohen who stands there to serve the Lord, your God, or to the judge that man shall die, and you shall abolish evil from Israel.
So you don't need to take my word for this, see for instance this entry in the Jewish Encyclopedia about the rebellious elder:
An elder who defies the authoritative rabbinic interpretation of the Mosaic Law. In the period when the Sanhedrin flourished this was a capital offense, punishable by strangulation (Sanh. xi. 1). This is based on Deut. xvii. 8-13, and according to the Talmud refers not to an ordinary man who refuses to abide by the decision of the priest or the judge, but to a regular ordained rabbi, or a judge, or an elder over the age of forty, or one of the twenty-three jurists constituting the minor Sanhedrin of a city or town. If such a judge dared to defy the decision of a majority of the major Sanhedrin, he became liable to the penalty of strangulation. R. Meïr, however, would convict only an elder whose opposition concerned a criminal act which, if committed unintentionally, would entail a sin-offering, or, committed intentionally, would be punished with excision (=כרת). According to R. Judah, the elder could be convicted only of a schismatic decision concerning a law which had its origin in Scripture, but the interpretation of which was left to the Soferim.