He likely was too ashamed of his history.
Tom Riddle did mention a reason why he didn’t want to get into politics. Though he certainly had a tendency to be untruthful at that stage in his life, it’s worth at least considering what he said about it himself, since it may be at least partially true. When Slughorn suggested that Tom Riddle get into politics, he told Slughorn he didn’t think politics would suit him, saying he had the wrong kind of background.
“– I confidently expect you to rise to Minister for Magic within twenty years. Fifteen, if you keep sending me pineapple. I have excellent contacts at the Ministry.’
Tom Riddle merely smiled as the others laughed again. Harry noticed that he was by no means the eldest of the group of boys, but that they all seemed to look to him as their leader.
‘I don’t know that politics would suit me, sir,’ he said when the laughter had died away. ‘I don’t have the right kind of background, for one thing.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
While he may have also had other reasons, it seems likely that what he said was at least partially true. We know he despised his name and Muggle heritage, since he decided early on to change his name.
“TOM MARVOLO RIDDLE
Then he waved the wand once, and the letters of his name re-arranged themselves:
I AM LORD VOLDEMORT
‘You see?’ he whispered. ‘It was a name I was already using at Hogwarts, to my most intimate friends only, of course. You think I was going to use my filthy Muggle father’s name for ever? I, in whose veins runs the blood of Salazar Slytherin himself, through my mother’s side? I, keep the name of a foul, common Muggle, who abandoned me even before I was born, just because he found out his wife was a witch? No, Harry. I fashioned myself a new name, a name I knew wizards everywhere would one day fear to speak, when I had become the greatest sorcerer in the world!”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)
Going into politics right after Hogwarts would ensure that everyone would recognize him as Tom Riddle even if he still changed his name, because he’d be visible to the wizarding world. By not going into politics, he was able to disappear from public view, and reappear later as Lord Voldemort without many people remembering who he once was.
“But before they were sure beyond doubt that the cup and the locket were both gone, the assistant who had worked at Borgin and Burkes, the young man who had visited Hepzibah so regularly and charmed her so well, had resigned his post and vanished. His superiors had no idea where he had gone; they were as surprised as anyone at his disappearance. And that was the last that was seen or heard of Tom Riddle for a very long time.”
- Harry Potter and the Half-Blood Prince, Chapter 20 (Lord Voldemort’s Request)
It seems likely he wouldn’t have wanted a career in politics because he was ashamed of his background and history, so wouldn’t want to enter into public life with that history.
As Lord Voldemort, he remained uninterested.
Even after he’d fully become Lord Voldemort, and could easily have taken the position of Minister of Magic without concern of his past, he didn’t want it. He chose instead to install a puppet Minister to run the Ministry, leaving him free to do other things.
“Why didn’t Voldemort declare himself Minister for Magic?’ asked Ron.
‘He doesn’t need to, Ron. Effectively he is the Minister, but why should he sit behind a desk at the Ministry? His puppet, Thicknesse, is taking care of everyday business, leaving Voldemort free to extend his power beyond the Ministry.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
Then as well, he wasn’t interested in having an official political post, although he easily could have at that point.