Why is this voting system not secure? Why not have each person simply treated as a counterparty to an off-the-record chat?


People show up in groups and show an ID which is cross-referenced against a central database to prevent Sybil attacks.
They have their apps generate a set of random numbers by each committing to one or more values and then revealing their values, combining them to form some set of random numbers.
Based on this, they each receive a provably random series of off-the-record MAC keys to use for upcoming elections onto their device, without the issuer knowing anything about the device, e.g. scan QR codes.
Then in each election, they vote once using Off-The-Record Messaging and have all the votes stored in a Merkle tree?


This scheme has the following properties:


Verifiability - seeing one’s own Merkle branch is all that’s needed to verify the vote was counted correctly. Seeing the whole Merkle tree one can show that a certain % of issued tokens were used and used only once.
Unlinkability - just like in OTR chat, once you sent the message, the zero-knowledge proof has been conveyed but now no one can prove how you voted.


Is there a flaw I missed? Why can’t we do voting like this?
Unfortunately, most of the reasons that e-voting is not popular have  nothing to do with the integrity of the underlying mathematics.  This makes the question slight moot on a cryptography forum, but the following un-cryptographic answer still entirely relevant.  Unless you solve the following (non exhaustive) problems:-


It's a  requirement of law enforcement (and thus oddly necessary for the protection of democracy) that you have to be able to check how people voted. Just proving that someone voted once is insufficient. Zero knowledge by definition can't be evidence in a proceeding.
Ella points out the conundrum that if point 1 is satisfied and voter behaviours can be verified, it can encourage more vote fraud by people selling their votes which can then be machine verified as bought by the orchestrator. Could this lead to an alt-Trustpilot system emerging of "quality" voters ranked by cost and reliability?
You also have to be able to prove who hasn't voted so that they can be prosecuted in certain jurisdictions /Australia. This is probably easier to solve than point 1, but clearly the issue of tokens does not guarantee a cast ballot. But such proof gets complicated again if run in parallel with traditional paper schemes.
Your requirement for ID is a H U G E problem.  25% of African-American voting-age citizens lack a US issued photo ID. My mum and dad don't have any either. Ginswich points out that several of these points share the common theme of  voter identification /authentication. This speaks to the broader issue of whether we want a society where everyone must carry "papers".
Can I spoil my vote?  I reserve the right to draw a squirrel on my voting slip or have an option for non of the above.  This must be the easiest to solve with the addition of a dummy candidate.
The bad guys will just hack the phones. So the Merkle tree remains intact collating the votes, it's just that you can set the app to vote for Hillary all the time whilst swearing to you that it registered a vote for the other one. Or play a tune instead on the central collation point.
What are these apps you mention?  I'm blind so I can't easily use a phone and therefore automatically technologically disenfranchised.  Although progress on accessibility is being made, how would the blind and /or deaf interact with a QR code?
And if I don't have a phone (my granny doesn't) can I borrow someone else's to vote with legally? Can multiple vote tokens be issued to the same device if sharing?
My daughter has three phones. 
What is the relationship between the contents of the Merkle tree and any cast/spoiled paper ballots, or do you plan to go 100% paperless? That would be quite a societal change.


Before the deluge of up votes, please consider that some cryptography is fine academically on a black board, but it often has to operate in the real political and social world.  A broader view has to be taken of cryptography and it's role in   society.

Update:

Regarding voter anonymity in parts #1 and #3, and comments. The following is the back of the current UK Parliamentary Election ballot paper. You may spot the unique bar and QR codes. The bar code is over 70mm long so it's not really a secret.   It's not a problem as long as the voting remains paper based.