This doesn't seem too difficult, and I don't really see how block-chain would contribute meaningfully. It should be possible to provide a solution at least as secure as paper voting, though it requires a similar level of trust in the organizations running it.
Once the gatekeeper has received your encrypted vote, it marks on the registry that you have voted. This is similar to the poll worker at a paper voting station - they know who you are and that you have voted, and they are in possession of your ballot, but can't see what your vote actually is because it is locked in a box (or encrypted with a key they do not have in this case). The gatekeeper, just like the poll workers manning an in person voting location, is responsible for transmitting your vote to the central location.
The gatekeeper signs the vote with its private key before sending it on to the tabulator so that all votes received by the tabulator can be confirmed to have come from a gatekeeper, reducing the likelihood of fake votes. The tabulator would also, of course, only accept votes over encrypted connections from known gatekeeper IP addresses. Any other best practice to ensure trust would also be taken, of course.
Once the vote is sent to the counting facility to be counted, this proceeds not much differently than current electronic voting. The vote is decrypted and tallied via software, though unlike current electronic voting machines, the code would be fully open source. Again, non- and bi-partisan observers would be allowed to validate that the process is proceeding honestly - with experts overseeing the setup, configuration and installation, as well as the operation, of the computer equipment.
So, you have to trust the people validating the hardware and software and those auditing the code - which, remember, are both non-partisan and bi-partisan to ensure fairness - much as you have to trust the poll workers and the vote counters in current paper elections. Is it fool-proof? No, but it is at least as fool-proof as paper or mail-in ballots.
There could be multiple gatekeepers and which one you use could be based on your district or other factors. The gatekeepers would have to delay transmission of a vote to the central authority until it can confirm that that voter hasn't voted (say at another gateway or in person), so regular updates to the voter rolls would have to be maintained, again, not much different than current. Standard transaction queuing software/algorithms could be used to ensure that service disruptions don't lose any votes, similar to how banks ensure the safety of transactions.
Once you vote, you could print out a confirmation page, but the information it showed would be minimal - maybe a transaction ID and a date/time, but even that might be too specific - say if the tabulator received a vote from a specific gatekeeper at that specific time (or, more accurately, a small batch of queued votes shortly after that once the gatekeeper had confirmed the uniqueness of your vote after a frequent voter roll sync). The information here is adjustable. Considering that you don't get anything other than an "I voted" sticker from a regular polling location, this could simply be a simple "we've received your vote!" message.
For redundancy, the gatekeepers could keep local copies of the votes they received, simply dissociated from any identifying info (including IP addresses stored in web-server logs). This is optional, but could be useful in double checking vote counts. I should state, here, that this is not a fully formed idea, and so some of the details will need refining by experts in such things, hence why this is an optional item as well as why some of the details of the audits are not explicit.
Further, I am concerned that there is some flaw in this plan that I am missing - experts have spent way more time on this problem than I have and have mostly decided that secure and anonymous voting is not possible outside of paper ballots, so I am likely falling into the old security trap of "I don't see a flaw, so it must be fool-proof!". But, I humbly post it here anyway as I haven't found a similar plan proposed and am curious to see what flaws there are.
Lastly, concerning the argument that "people don't understand the technology and so won't trust it", I don't recognize that as valid because people use technology they don't understand all the time. Most folks don't understand public key encryption, and yet they bank online and purchase online using it every day. Current elections use fancy machines to count and tabulate votes and no one really blinks at that because it is safely buried behind the scenes (where they can ignorantly assume that every vote is counted by hand). With the vetted by trusted experts (regardless of your political leanings), this should be no different.