Yes, Sidious did let Windu beat him The number one reason why many Star Wars fans believe that Darth Sidious did let Mace Windu defeat him is that it was the perfect way to finalise Anakin's turn to the Dark Side.
It's implied that this isn't feigning and Palpatine is actually fighting to win. In the book its implied that he allows Mace to move him and make certain moves, however this is a not uncommon dueling tactic, waiting for openings and surprising an opponent.
With the Jedi and Sith seeming philosophically the same to Anakin, he inevitably chose the side that at least might save his wife. Anakin dismembered Windu, allowing Palpatine to drop his façade of weakness and assault the Jedi Master with lightning once more, killing him.
Originally Answered: Why didn't Palpatine just stop his force lightning when Mace Windu deflected it back at him? It was all part of his manipulation of Anakin and the Senate. He needed Anakin to think Mace Windu was executing a helpless prisoner, and he needed the Senate to see evidence that the Jedi were traitors.
Vader relies on the complex life support system of his suit to survive and it happens to be very vulnerable to electrical discharges. So, if Vader ever had unleashed Force lightning, there's a good chance it would've caused his suit to short circuit, thus, killing him.
The showdown on Coruscant was a confrontation that took place in the Supreme Chancellor's Suite of the Galactic Senate in 19 BBY. It began after Jedi Knight Anakin Skywalker informed Master Mace Windu that Supreme Chancellor Palpatine had revealed his identity as a Sith Lord to him.
This isn't to say that Mace Windu isn't powerful when it comes to the force. He's on par with Darth Maul and is more powerful than Qui-Gon Jinn. But, Obi-Wan is still the stronger Force user which makes him a powerful combatant.
Yoda feared that defeating Sidious would somehow make Anakin stronger. He knew that the only way to victory was to kill both the apprentice and the master at the same time.
Giving in to his feelings, Anakin cuts off Mace Windu's hand, giving Palpatine the opportunity to kill him. Anakin regrets this almost instantly, but the thought of Padme was stronger than his remorse This is the moment Anakin officially became Darth Vader and led to serious issues in the Skywalker Saga.
In short, he thought that the Jedi were becoming too subservient to the Republic and the Senate. That was on particular display in "Choices," when he and Master Windu were working as partners.
Sidious himself felt that he and his peers had outgrown the use of lightsabers, and only continued carrying them to mock the Jedi. He ultimately viewed his own lightsabers as little more than an affectation, and rarely wielded them in combat.
This information comes to light form the comic series Darth Vader: Dark Lord of the Sith by Charles Soule and Giuseppe Camuncoli, in particular issue #7. Palpatine admits to Darth Vader that the Jedi he actually feared most was none other than Jocasta Nu.
And while Yoda and Palpatine both worked with the Senate, they treated each other with the utmost respect. While Palpatine plotted in the shadows and planned his Galactic Empire, he underestimated Yoda. And even when Yoda discovered Palpatine's true nature, Yoda underestimated his power.
If the two spared as Jedi, Dooku would probably win the match because he was the better all-around duelist. But if this issue's fight had continued with Windu as a Sith, Windu would have been able to turn the tables.
Mace windu clear winner 9/10 times. His use of vapad would make the fight play out similair to windu vs palpatine. He would use the seething rage pouring out of maul against him, and as mauls force power was so much weaker than windu's it would be an undisputed win for windu.
Darth Sidious / Emperor Palpatine
Darth Sidious was one of the destructive maniacal Sith lords and the most powerful character in Star Wars.
We go for the more simple answer, further considering the details of the question itself. If Anakin didn't go into that chamber as the Jedi confronted Palpatine, then Mace Windu would have defeated Palpatine, killing him.
The reason Vader never successfully overthrew the Emperor wasn't from lack of ambition, but more from a lack of allies. Even in Revenge Of The Sith Anakin planned to surpass Palpatine by telling Padme that they could overthrow the Chancellor and rule the galaxy together.
The canonical comic Star Wars: Darth Vader #7 confirms the younglings' deaths caused Anakin grief, self-loathing, hatred and pain, fueling his descent even further into the dark side. Those negative emotions also power the dark side, meaning Anakin likely felt stronger as he killed the younglings.
Jedi Masters Mace Windu, Kit Fisto, Saesee Tiin, and Agen Kolar confronted Chancellor Palpatine upon learning of his secret identity as a Sith Lord.
The end of George Lucas' prequel trilogy instead proves that Palpatine's face changes as part of his plan to manipulate Anakin, with Palpatine using the Force to hide his true visage up until that moment.
In Luke Skywalker's confrontation with Palpatine during The Return of the Jedi, he tells the Emperor that his overconfidence is his weakness. That turns out to be true as Palpatine takes too long trying to kill the young Jedi, which gives Darth Vader...