If The Ministry has the power to detect what specific spells have been used, how didn't they know that Morfin's memory had been edited?

Question

Chapter 17 of Half-Blood Prince gives us this explanation for how Morfin Gaunt was framed for the murder of the Riddle family:

Voldemort Stupefied his uncle, took his wand, and proceeded across the valley to ‘the big house over the way.’ There he murdered the Muggle man who had abandoned his witch mother, and, for good measure, his Muggle grandparents [...]. Then he returned to the Gaunt hovel, performed the complex bit of magic that would implant a false memory in his uncle’s mind, laid Morfin’s wand beside its unconscious owner, pocketed the ancient ring he wore, and departed.

Harry objects to this story, claiming:

He [Tom Riddle] was underage at the time, wasn’t he? I thought they [The Ministry] could detect underage magic!”

Dumbledore explains this by claiming:

You are quite right — they can detect magic, but not the perpetrator:

This effectively ends the conversation. However, I fail to see how it resolves the problem. My first objection is that if The Ministry specifically detects magic near the underaged, then they should know that somebody other than Morfin was present, raising questions about the events. My second is much more substantial. Chapter 2 of Chamber of Secrets leaves no doubt that when The Ministry detects underaged magic, they can detect the specific spells used:

We have received intelligence that a Hover Charm was used at your place of residence this evening at twelve minutes past nine.

and a very similar case can be found in Order of the Phoenix, differing only in that it specifically identifies Harry as the caster:

We have received intelligence that you performed the Patronus Charm...

So, if it is the case that The Ministry can detect the specific spells that were cast in an area, then how did they not know about the stunning spell and memory modification that Tom Riddle performed, calling in to question Morfin's confessions?

As revealed while discussing the current answers, this situation becomes even worse if we read in to the fact that the letters to Harry gave the time of the spell castings. This means that only does The Ministry apparently know what spells were used, they know the order that they were cast in, up to and including (if Dumbledore's guess is accurate) the time gap between the final murder and the use of the memory-changing spell. I can't quite see Morfin's confession being taken serious in a world where somebody wrote the letter Dear Mr Tom Riddle, We have received intelligence that a Stunning Spell, Avada Kedavra, Avada Kedavra, Avada Kedavra, and, after a break of a few minutes, a Memory Charm was used at....

Answer 1

Because Morfin confessed that he killed the Muggles

There was no need to check for memory alterations when the accused (who as known to attack the village muggles and the Ministry officials) himself confessed killing Tom Riddle Sr and his family.

They also 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 ...
_{Half Blood Prince: Chapter 17 - A Sluggish Memory}

Later,

“And Morfin never realized he hadn’t done it?” “Never,” said Dumbledore. “He gave, as I say, a full and boastful confession.”

“But he had this real memory in him all the time!”
_{Half Blood Prince: Chapter 17 - A Sluggish Memory}

The murder weapon (wand) showed that it was used to kill the Riddle family, the owner of the wand confessed, so there was no need to check anything further.

---

Also, even if we consider that the Memory Alteration charm was detected by the Ministry, it would simply mean that Morfin's wand was used; the charm would neither indicate if was Morfin or someone else casting the spell nor would it show the spell's target.

“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. “So if you’re underage and you do magic inside an adult witch or wizard’s house, the Ministry won’t know?”

“They will certainly be unable to tell who performed the magic,” said Dumbledore, smiling slightly at the look of great indignation on Harry’s face. “They rely on witch and wizard parents to enforce their offspring’s obedience while within their walls.”...
_{Half Blood Prince: Chapter 17 - A Sluggish Memory}

To add, the Ministry had no intention of using Legilimens (I doubt whether that actually had strong Legilimens like Dumbledore or Snape at their disposal) on a confessed crime. And from his past actions, Morfin was an easy target to nail. It was only Dumbledore who thought that Morfin was wrongfully accused and investigated it further:

“Yes, but it took a great deal of skilled Legilimency to coax it out of him,” said Dumbledore, “and why should anybody delve further into Morfin’s mind when he had already confessed to the crime? ...
_{Half Blood Prince: Chapter 17 - A Sluggish Memory}

Answer 2

The Trace, the Trace!" said Mad-Eye impatiently. "The charm that detects magical activity around under-seventeens, the way the Ministry finds out about underage magic! If you, or anyone around you, casts a spell to get you out of here, Thicknesse is going to know about it, and so will the Death Eaters."

So the Ministry only know when magic near a underage wizard is done not who actually do the magic.

In a home with adults wizards the trace is gonna detect the spells done by adults so the Ministry probably ignora that magic detected by the Trace.

Morfin was 43 years old when his nephew altered his memory so the Ministry detected the spell but they ignored it like all the magic done in public place or in place like the Burrow.