Siddurim have only been printed for the last few centuries. Before that they were handwritten and very expensive. Indeed, prior to the invention of typesetters and pulp paper, all sefarim were very expensive. From records of early printers, it appears that books that were heavily used frequently have no surviving copies (and with pre-1500, even fewer existed).
If there is a record of someone introducing a tefilah, that answers your question. If it suddenly appears, that allows you to date it. Consider dating the origin of the prayers for the Israeli government and the IDF.
Do the temanim say “Modeh ani”? Does anyone not say it? In what early siddurim is it absent? Is it attributed by contemporaries to an author (as would be the case if it is recent, meaning the last 1500 years).