Your comment prompted me to double check something. I learned a little bit of Icelandic and þ is used for th as in ‘thin’, while ð (eth) is used for ‘this/the’.

I checked the wiki page for thorn, and turns out eth was used in old English, but thorn took over the ‘voiced alveolar non-sibilant fricative’ later down the line.

Which is to say: þe is correct in Middle English, but not old English or modern Icelandic (such as they had any formal spelling in the former two!)