I wrote:
Date input requires a full date. Partial date input is not handled by Calc.
That is not quite right. However, you need to add something by way of separators or unambiguous data (month names) to have the input conforming to a date format Calc will take to be a date (use a “date acceptane pattern”). See comments from erAck and Opaque for details.
Cell formatting only affects how the content is displayed, not how input is interpreted. Also, even though you instructed Calc to interpret as dates, it can’t read the dates and falls back to making text cells.
Using the DATEVALUE() function may help, but I suspect that it doesn’t add much to Calc’s date recognition. You will get an error for content not handled as a date, which may be useful.
Probably better: Use space as separation character in Text to columns, and keep default column type. This will give two columns of numbers, one for month and one for year. You can sort on those columns as they are (first sort by year then by month). If you want a proper date, use the DATE() function to assemble year + month + 1 to a proper date value (which can also be sorted as you need).