If Hagrid couldn't take a portkey because they were being watched, how could they take a portkey to Grimmauld Place?

Question

In Order of the Phoenix there are multiple discussions about travel methods being watched.

Chapter Twenty:

“A month?” said Ron, as though he had never heard of a journey lasting such a ridiculously long time. “But — why couldn’t you just grab a Portkey or something?”

There was an odd expression in Hagrid’s unobscured eye as he squinted at Ron; it was almost pitying.

“We’re bein’ watched, Ron,” he said gruffly.

“What d’you mean?”

“Yeh don’ understand,” said Hagrid. “The Ministry’s keepin’ an eye on Dumbledore an’ anyone they reckon’s in league with him, an’ —”

Chapter Twenty-Two:

“How’re we going?” asked Fred, looking shaken. “Floo powder?”

“No,” said Dumbledore, “Floo powder is not safe at the moment, the Network is being watched. You will be taking a Portkey.” He indicated the old kettle lying innocently on his desk. “We are just waiting for Phineas Nigellus to report back... I wish to be sure that the coast is clear before sending you —”

If Hagrid couldn't take a portkey because they were being watched, why were Harry and the Weasleys able to take a portkey? Dumbledore was still clearly worried about being watched, as that is the very reason he gives for why they can't use the Floo Network.

Perhaps relevant is another discussion earlier in the book where potential travel via portkey is discussed:

Chapter Three:

“How’re we getting — wherever we’re going?” Harry asked.

“Brooms,” said Lupin. “Only way. You’re too young to Apparate, they’ll be watching the Floo Network, and it’s more than our life’s worth to set up an unauthorized Portkey.”

Answer 1

Because Hagrid using a portkey to suddenly disappear for many days would definitely be suspicious.

Portkeys seem to be different from the Floo network in that their use cannot be monitored, but the creation of the portkey somehow can. But, if they used the portkey in a way that roused suspicion, it would still be a problem.

Hagrid's mission was to try to form an alliance between the giants and the Order. They knew beforehand that he would not be able to do this in a couple hours, rather it would take days. Anyone keeping tabs on the Order would definitely become suspicious if Hagrid suddenly up and disappeared from his job for many days without any explanation. Also, his destination, the giants' home, was not magically protected. If someone became suspicious, it would be possible to discover his whereabouts, which would reveal their plans, or they could spy through magical means. Therefore, he put on a show of going on vacation, making sure there were many witnesses that could confirm that he was in fact traveling.

Also of note, Hagrid cannot use magic on his own really, and portkeys are normally one-way with the tri-wizard cup being an exception, possibly due to double portkey charms Why was the Triwizard cup portkey in Goblet of Fire two-way? If they had not thought to do this, Hagrid would be trapped with the giants on a one-way trip. I imagine also that there may have been worries that if it were a two-way portkey, if a giant touched the key, that's a much bigger problem than if a random human touches it.

Meanwhile, when they used the portkey to go to Grimmauld Place, it was to see Arthur who had been attacked at the ministry. It was no secret that Arthur had been attacked, and therefore using it would reveal nothing of their plans. Grimmauld Place was also magically protected, and so as long as the portkey itself was protected, they could not be followed or discovered.

As for how this portkey was created without any problems with the Ministry, it is never said, but presumably Dumbledore found some way to finagle it, perhaps because as headmaster of Hogwarts and highly respected in the Wizarding community, he has enough political power to do so without repercussion. As further evidence to support this theory, after the battle at the ministry, Dumbledore creates another portkey, in front of Fudge no less, to bring Harry back to Hogwarts.

Answer 2

When it comes to portkeys, I don't think the issue of being watched centers around magical tracking methods. Indeed, in the Chapter Twenty-Two quote Dumbledore explicitly contrasts portkeys to the Floo Network in that the former is not being tracked. Similarly, in the Chapter Three quote Lupin does not reject portkeys because they are being tracked, while he does reject the Floo Network for that reason.

Rather, I think Hagrid's concern is that they are being physically followed. If Hagrid picks up and old can and is whisked away, the Ministry spies will see that he just took a portkey. This is in fact borne out by what Hagrid continues to say:

“— so we’re not hard ter follow. We was pretendin’ we was goin’ on holiday together, so we got inter France an’ we made like we was headin’ fer where Olympe’s school is, ’cause we knew we was bein’ tailed by someone from the Ministry. We had to go slow, ’cause I’m not really s’posed ter use magic an’ we knew the Ministry’d be lookin’ fer a reason ter run us in. But we managed ter give the berk tailin’ us the slip round abou’ Dee-John —”

However, when it came to traveling to Grimmauld Place this was not an issue. Even if Dumbledore and his followers are being watched, the spies can't see what happens in the confines of Dumbledore's office. The only person inside of Hogwarts that might have posed a problem was Umbridge, and Dumbledore specifically planned around this:

“It is Fawkes’s warning,” said Dumbledore, catching the feather as it fell. “She must know you’re out of your beds... Minerva, go and head her off — tell her any story —”

Once they took the portkey they could not be tracked so everything would be fine. At worst it would be a case of a few students leaving school for the holidays a little early. But there wouldn't be any evidence of Dumbledore or his followers doing anything subversive or illegal.

Of course you might then ask that if it was so simple to take a portkey from Dumbledore's office where no one could see what was happening, why didn't Hagrid just take a portkey from Dumbledore's office as well? The answer to that would be that it wouldn't accomplish much. The spies might not be able to see what actually happened in the office, but they would surely see Hagrid entering the castle, if not the office itself. When he would fail to come back out it would be obvious that he had gone somewhere. The prime suspicion would probably be a portkey, and even if they couldn't prove that they would still be able to figure out that Hagrid was performing a secret mission for Dumbledore.

Thus, the best option for Hagrid was to make it seem like he was just going on a regular journey, and hope to lose the tail at some point (which he did) and then make it to the giants.