Dumbledore describes why they believed it was Morfin:
'The Ministry... knew that a convicted Muggle-hater lived across the valley from the Riddle house, a Muggle-hater who had already been imprisoned once for attacking one of the murdered people.
'So the Ministry called upon Morfin. They did not need to question him, to use Veritaserum or Legilimency. He admitted to the murder on the spot, giving details only the murderer could know. He was proud, he said, to have killed the Muggles, had been awaiting his chance all these years. He handed over his wand, which was proved at once to have been used to kill the Riddles.'
So they had means, motive, and opportunity, along with a confession. All without any real investigation. Even Morfin believed he had been the one to kill them:
[Harry:] 'And Morfin never realised he hadn't done it?'
'Never,' said Dumbledore. 'He gave, as I say, a full and boastful confession.'
When Harry asked about Tom being under-age, Dumbledore just says they can't tell who cast it:
'But how come the Ministry didn't realise that Voldemort had done all that to Morfin?' Harry asked angrily. 'He was under age at the time, wasn't he? I thought they could detect under-age magic!'
'You are quite right - they can detect magic, but not the perpetrator: you will remember that you were blamed by the Ministry for the Hover Charm that was, in fact, cast by -'
'Dobby,' growled Harry; this injustice still rankled.
That doesn't explain why they glossed over Tom's trace, he was 16 so it should have registered. Perhaps because he was a Prefect moving up to Head Boy. Maybe they did question him, and he said he came to find his Uncle and just happened to be nearby when Morfin killed them - that seems to me the likeliest.
All quotes from HBP, ch. 17 (pp.343-344, British first ed. hardcover)
