Given a string of inputs, it is possible to execute it using :normal!: :normal! ihello will go into insert mode and write the text 'hello'.
When mapping keys to expressions we can use a similar, albeit incompatible syntax. In Lua, we could, for example, define
vim.keymap.set('n', 's', '<c-a>y')
Given such a string, '<c-a>y', how would we execute it imperatively from within Lua/Vimscript without a key binding, similar to what is possible with :normal!?
"cmd" notation is not a replacement but alternative. Usually, the RHS of a mapping is processed as if (well, almost) typed. However, any "cmd"-thing is being processed much like as a script.
Therefore, you are supposed to strip "cmd" and "cr" off and pass it onto
execute, notnormal. Of course, after make sure that there is no other<key>notation left.