But it seems you also modified either tab-line-tabs-buffer-list-function,
or tab-line-tabs-buffer-group-sort-function, or both of them. Is this true?
Yes to tab-line-tabs-buffer-group-sort-function, no to tab-line-tabs-buffer-list-function, yes also to tab-line-tabs-buffer-group-function. Setup is below:
(defvar ++buffer-group-table
(make-hash-table :test 'equal))
;; I manually populate ++buffer-group-table
(defun ++tab-line-buffer-group (buffer)
(gethash (buffer-name buffer) ++buffer-group-table
(with-current-buffer buffer
(concat (symbol-name major-mode)
":"
(and buffer-file-name
(file-name-directory buffer-file-name))))))
(defun ++buffer-name< (a b)
(string< (buffer-name a) (buffer-name b)))
(setq tab-line-tabs-function #'tab-line-tabs-buffer-groups
tab-line-tabs-buffer-group-function #'++tab-line-buffer-group
tab-line-tabs-buffer-group-sort-function #'++buffer-name<
tab-line-switch-cycling t)