I've read that it was in order to prove he didn't take money from the Treasury in order to pay the bribe, but could it also be attributed to his penchant for honesty and values?
Also, why was he still asked who he'd vote for during the 1800 election, despite still recovering from this scandal? By all accounts, his political career should have ended then and there.
Edit: The Reynolds Pamphlet.
Hi! So, I am by no means a historian, and have a some-what limited understanding of the affair, but I do believe I have some reliable sources, but I still ask you to please, please, please take this answer with a grain of salt. As far as I know, Hamilton first met with a few members of the cabinet (including Jefferson, Madison, and Burr) who had noticed that he was sending checks to James Reynolds, and had told them of the affair to prove his innocence. However, afterwards, he believed that his confidantes would eventually tell people of the affair, and he decided that it would be better for the American public to find out from him. So, he published The Reynolds Pamphlet.
As to the second part of your question, in the election of 1800, Jefferson and Burr tied with 73 votes each in the electoral college, and they, who were previously running partners, were turned against each other. In the event of a tie, the constitution states that the decision should be left up to the House of Representatives. Now, James A. Bayard wrote to Hamilton about who he would vote for in this exact event earlier in the election when it seemed certain that Jefferson and Burr would be taken to the House of Representatives, and Hamilton published his response supporting Jefferson. Hamilton does say in his response to Bayard that he believes the Federalists in the house may prefer Burr, which may effect Burr's action's later. Now, as to if this actually effected any house members' votes is unknown, but when Jefferson beat Burr (10 votes to 4), Burr took Hamilton's endorsement of Jefferson as an insult and in his outrage challenged Hamilton to a duel. So, to answer your question, the American public as a whole did not ask Hamilton for his opinion, and why would they? You are completely correct in saying his political career was in shambles after the events the followed Washington's retirement eg. The Reynolds Pamphlet, the Adams Administration.
This is my first time answering on this subreddit and one of my first times seriously commenting on Reddit as a whole, so please, forgive me! But, this is my understanding of the matter and I hope I answered your question well.
EDIT: Formating