How to mark org agenda item as completed on particular day?

Question

Sometimes I fall behind in my org-agenda maintenance and have to go back a few days to mark completed tasks. In 90% of cases, I want to mark the task as completed on the day it was due, but org-agenda-todo always uses the current day and time.

What is the easiest way to mark agenda items as completed on the day currently being viewed in the agenda window?

Update: I attempted to adapt the answer to the linked question; ignoring the ugliness of my code, the date seems to be obtained, but org-agenda-todo ignores org-current-effective-time:

(defun org-agenda-todo-view-date (&optional arg)
  (interactive "P")
  (cl-letf* ((org-read-date-prefer-future nil)
             (my-current-time (org-read-date nil t (mapconcat 'number-to-string (append (last (calendar-gregorian-from-absolute org-starting-day)) (butlast (calendar-gregorian-from-absolute org-starting-day))) "-")))
             ((symbol-function #'org-current-effective-time)
              #'(lambda () my-current-time)))
    (org-agenda-todo arg)))

Answer 1

This is similar to another question where the goal was to prompt for the date. Here you just want to use the date under the cursor. We can basically use the same answer, but we get the date in a different way. Instead of prompting with org-read-date, we use org-get-cursor-date. With this the CLOSED property and any logbook entries will be made on whatever date in the agenda you happen to be on when you call org-agenda-todo-view-date.

(require 'cl-lib)

(defun org-agenda-todo-view-date (&optional arg)
  (interactive "P")
  (let ((my-current-time (org-get-cursor-date)))
    (cl-letf* (((symbol-function #'org-current-effective-time)
                (lambda () my-current-time))
               ((symbol-function #'org-today)
                (lambda () (time-to-days (org-current-effective-time)))))
      (let ((current-prefix-arg arg))
        (call-interactively 'org-agenda-todo)))))

Note that the function responsible for updating schedules, org-auto-repeat-maybe, uses org-today to get the date instead of org-current-effective-time (which org-today apparently does not call) so we modify it as well to match.