Why did Admiral Piett allow the Shuttle Tydirium to pass?

Question

Why would Admiral Piett allow a shuttle stolen by the Rebel Alliance to pass?

In Return of the Jedi, the Rebel Alliance use a stolen Lambda-Class Shuttle (Tydirium) to infiltrate the Imperial held moon Endor.

"We have stolen a small Imperial shuttle. Disguised as a cargo ship, and using a secret Imperial code, a strike team will land on the moon and deactivate the shield generator." ―General Crix Madine

It appears that the Imperials have some mechanism of identifying a ship. In Star Wars: Rebels, they are able to identify the Ghost rather than a generic VCX-100 light freighter.

Furthermore, we know the Admiral Correctly Identifies Tydirium:

"Shuttle Tydirium, what is your cargo and destination?" ―Admiral Firmus Piett

In canon, it is unclear whether the Empire was aware that Tydirium was stolen. (Legends there is a high speed pursuit ended with a Rogue Squadron rescue and a hyperspace jump).

I find it hard to believe that the Empire did not notice a valuable Lambda Class shuttle went missing; let alone, one carrying access codes to Endor. If Tydirium was reported as stolen by Rebels; why would they allow it to pass through to Endor:

Darth Vader: Do they have a code clearance?

Admiral Piett: It's an older code, sir, but it checks out. I was about to clear them.

There are a few explanations:

Admiral Piett was in on the Emperor's Plan

I do not see any evidence to support nor debunk this completely. Piett was given direct orders concerning the Rebel Fleet at Endor:

Admiral Piett: I have my orders from the Emperor himself. He has something special planned for them. We only need to keep them from escaping.

Admiral Piett had his own plan

There is no evidence to support this. My theory was that Piett saw the stolen shuttle and figured the Rebels would be easier to capture on Endor rather than while in space and still capable of jumping to hyperspace. His plan is put on hold when Vader tells Piett that HE would deal with them.

Admiral Piett: Shall I hold them?

Darth Vader: No. Leave them to me. I will deal with them myself.

The shuttle's status was not known

There is no evidence to say the canon Imperials knew that the shuttle was stolen. I find this hard to believe as the Lambda class shuttle was usually assigned to an officer or crew for high valued cargo/personnel. I also find it unlikely that a shuttle carrying secret Imperial codes to the Highly secure Endor Base/Death Star II would be left unchecked for more than a few hours. The codes being in Tydirium's computer is from Legends.

Answer 1

There's no good reason to assume that what Piett says to Vader isn't the absolute and unvarnished truth, something that's borne out by the film's various novelisations.

“It’s an older code, but it checks out,” Piett replied immediately. “I was about to clear them.”
There was no point in lying to the Lord of the Sith. He always knew if you lied; lies sang out to the Dark Lord.

_{^{Return of the Jedi: Official Novelisation}}

and

On the Executor’s bridge, Piett was starting to wonder about Vader’s interest in the shuttle. “Shall I hold them?” he asked.
“No,” Vader answered firmly. “Leave them to me. I will deal with them myself.”

_{^{Return of the Jedi: Junior Novelisation}}

The most recent (and fully canon) novelisation indicates that Piett was entirely unaware that anything was amiss, relying on his underlings to check and pass codes.

“Do they have code clearance?” demands Vader.
Piett doesn’t know, so he gestures to the flight controller, who always prayed he would never have to speak to Vader.
“It’s…an older code, sir, but it checks out,” he reports. “I—I was about to clear them.”
The controller braces himself. He knows what his fate might be if this is the wrong answer.

_{^{Return of the Jedi: Beware the Power of the Dark Side!}}

As to the question of why the Empire hadn't reported the shuttle missing, the novel Moving Target gives us all the info we need to answer that question.

“I’m fine. Where are you?”
“Kothlis,” Luke said, then peered at the hologram he was seeing of her. “Are you wearing an Imperial uniform?”
“I am,” Leia said. “Its previous owner won’t miss it—she’s space dust. Along with her Star Destroyer. Of which the shuttle Tydirium is all that remains.”

Answer 2

In his conversation with Darth Vader, Moff Jerjerrod --- the head of Death Star construction --- was terrified of the Emperor's displeasure at the project having fallen behind schedule. The shuttle was supposedly carrying supplies and personnel for the Endor base. With the Emperor and his hatchetman both on-site, the upper ranks were strongly motivated to avoid new delays, even on the Endor side of the operation.

Answer 3

The people in charge of construction of the Death Star II were under significant pressure to get on with it because it was behind schedule. It's not unreasonable to assume they therefore sent out panicked messages trying to pull in extra resources, including things like shuttles which normally didn't do a run to Endor, essentially sending out a call for anything they could get their hands on.

Piett was commander of the fleet whose primary reason for being there was to ambush the Rebels, and the construction wasn't his responsibility. The fleet would have known extra support had been ordered, but may not have been informed regarding who was supposed to show up with the additional support. Given the panic shown by Jerjerrod when Vader laid down the law about moving faster, even they might not known (or cared) who would show up in response to their request for more support.

There might have been a host of cargo ships and shuttles showing up in response to Jerjerrod pleading, begging, and calling in favours to bring in more personnel and supplies, and things like having the very latest up-to-date codes was probably not high on the priority list. If you look at real-life situations, that sort of thing isn't an unknown occurrence in normally lawful democracies, let alone in corrupt dictatorships: "Hey, look, this is a rush job, boss'll be upset if it's held up because of a problem with the paperwork, so cut me some slack here."