Cannot Go to Riften Glitch

It seems indeed that there is a bug related to thieves guild leadership:

If the player completes the main story arc before doing the side and improvement quests for Vex and Delvin, Brynjolf may not interact with the player and instead repeat that he is busy and that he will talk to you later. Consequently, becoming the Guildmaster will be rendered impossible. This may be a part of a bug where Karliah disappears and does not appear in either the Nightingale Hall or the Twilight Sepulcher. She does, however, return to the Snow Veil Sanctum where you first encounter her.

Possible Fix:

Run outside when finishing your first interaction with Brynjolf. You need to run outside as quick as possible by going to the ladder and exiting. You should see the quest update pop up reading: "Started Under New Management". Afterward, go back inside and proceed to Brynjolf. To talk to Brynjolf, let him go to the middle of the Cistern first, and wait. He then should start talking to you, finishing the quest. If this does not resolve the issue, it may be worth it to try attacking them, paying the reparations, and trying again.

Another possible fix is to get the quest about fixing debt from the guy at the Riften stables. The player will be told to talk to Sapphire. Instead, Sapphire should be sitting on the bridge just inside Riften and when approached will say "I don't have any business with you." or "We'll talk after the ceremony." Attack her 3 times, and this should cause her to start attacking the player notifying them that "Under New Management" has been completed and that the player should talk to Brynjolf. Upon entering The Ragged Flagon, the player will be required to pay 1000 gold to Vex as reparations for killing a fellow Thieves Guild member. After doing so, speak with Brynjolf and the questline SHOULD be completed.

As a last resort, you can use the console to fix this issue:

setStage TGLeadership 0

How to fix (2013 update)

I've tried the bugged saved game file I initially posted in my question with the current latest versions of the official patch, Unofficial Skyrim Patch (USKP) and Skyrim Script Extender (SKSE), and the saved game is now Aspect of Peryite-free.

I've configured SKSE to use its ClearInvalidRegistrations console command on every saved game load, which removes invalid scripts left running by uninstalled mods. This feature was introduced in v1.6.7 of SKSE:

add console command ClearInvalidRegistrations to remove invalid OnUpdate() registrations

This prevents orphaned OnUpdate() events and the resulting bloated/broken saves when removing certain mods. When applied to an already bloated save, it will stop growing further and instead shrink over time as the game processes all queued events. This may take hours depending on the amount of bloat.

To execute automatically after each reload, add this to \Data\SKSE\skse.ini:

[General]
ClearInvalidRegistrations=1

According* to a developer of the Unofficial Skyrim Patch (USKP), the underlying cause is because DA13 ('The Only Cure' quest) is never stopped.

A possible fix* for this:

In the meantime, load an affected save, then type stopquest DA13 into the console. That ought to put an end to that. Plus it’ll free up some unnecessarily leftover alias memory.

BTW, don’t use the console command on a save not bothered by it. Even though it may appear benign, console commands can have negative consequences you don’t know about.

If this doesn't work or if you experience issues like: NPCs not responding properly to conversation (not reply to you when you choose a certain conversation path - leaving you stuck), merchants not opening the buy/sell window, horses that can't be ridden, and other interaction issues, then the save game is most likely corrupted. Try the SKSE ClearInvalidRegistrations console command fix I've posted above, instead.

To prevent this from happening in the PC, get the latest version of the Unofficial Skyrim Patch. This has been fixed since version 1.1 of the USKP:

"The Only Cure" (DA13) was never stopped which would lead to the Aspect of Peryite randomly initiating conversation and being unable to exit.

^{* This info is from the old USKP bug tracker page at 16bugs.com, which is now not viewable. The new USKP bug tracker report page of this bug is at afkmods.iguanadons.net.}