Why did Dumbledore instruct Mrs. Figg to not tell Harry about the Wizarding World, even up to when he was ready to be admitted?

In Order of the Phoenix, chapter 2, Mrs. Figg reveals that she has been watching Harry for years, but was told by Dumbledore not to tell him about the Wizarding World:

“Why didn’t you tell me you’re a Squib?” Harry asked Mrs. Figg, panting with the effort to keep walking. “All those times I came round your house — why didn’t you say anything?”

“Dumbeldore’s orders. I was to keep an eye on you but not say anything, you were too young.”

I understand not telling him about everything when he's 4 or 5, but Harry visited her mere weeks before receiving his acceptance letters from Hogwarts. Keeping the information from him that late in the game seems really silly.

Heck, right when the Dursleys were getting streams of letters would have been a great time for Mrs. Figg to stop by and explain the whole situation, rather than making Hagrid track them down to the middle of nowhere.

Speaking of Hagrid, he expected Harry to already know all about the Wizarding World. Sure, the Dursleys weren't forthcoming about the information, but if Hagrid figured that Harry was old enough to learn form the Dursleys, and both Hagrid and Mrs. Figg were on orders from Dumbledore, then why didn't she tell Harry about magic and such?

Answer

She gives at least 1 reason in the rest of the quote.

dragging along the ground. ‘Why didn’t you tell me you’re a Squib, Mrs Figg?’ asked Harry, panting with the effort to keep walking. ‘All those times I came round your house – why didn’t you say anything?’

‘Dumbledore’s orders. I was to keep an eye on you but not say anything, you were too young. I’m sorry I gave you such a miserable time, Harry, but the Dursleys would never have let you come if they’d thought you enjoyed it. It wasn’t easy, you know ... but oh my word,’ she said tragically,

We see how Figg rationalized why she never broke her promise, because then the Dursley's would have stopped Harry from coming over.

Figg was a long term spy set up by Dumbledore. She was not known to Voldemort's followers nor to the Ministry of Magic itself, as we find out later in the book. This allowed her to stay close and keep tabs on Harry his whole life, without even his paranoid aunt and uncle finding out, moving, or limiting the access she had to him.

‘We have no record of any witch or wizard living in Little Whinging, other than Harry Potter,’ said Madam Bones at once. ‘That situation has always been closely monitored, given ... given past events.’

‘I’m a Squib,’ said Mrs Figg. ‘So you wouldn’t have me registered, would you?’

An entity not known to the ministry, near Harry is priceless. The Death Eaters' primary means of info comes from ministry leaks, and the ministry itself is untrustworthy (Fudge/Umbridge). Though to be fair having Mungdungus know about her is risky. :P

Dumbledore's reason for the promise fits in simply with how he handled Harry and everyone else why stopping Voldemort. Dumbledore always acted as he saw fit to best keep his assets alive, even if it meant that his assets were damaged in the process. Something Dumbledore apologizes to Harry for later in the series.

While it might have made sense for her to tell Harry after he got his letters, or even in the 4 years after, Dumbledore thinks long term. He knows that eventually Voldemort will come back, and Figg is his Ace in the hole for Harry protection.