You're partially correct (in a very specific way to be detailed in a second), and the people responding to you are also correct. Here's the connection:

Pre-Old English only had voiceless fricatives, so 's', 'f', and theta (Þ), but underwent a sound change just before we have attested Old English whereby all (singleton) fricatives became voiced between other voiced sounds. So, 's' > 'z', 'f' > 'v', and theta > ð. (Let me know if you're interested in how we know that this sound change happened even though we don't have attested physical evidence of it from English!) This is why we have alternations in Modern English like:

loaF : loaVes (originally it was hlaf : hlafas, with "f" in both words)

This means that, for much of attested Old English, "th" fricatives at the beginnings (and ends) of words would be voiceless, and "th" fricatives in the middle of words would be voiced (unless they were next to a voiceless sound, like the 'f' in soft, but I don't think there are any examples of that for 'th').

So, why do we have both voiced and voiceless fricatives initially, medially, and finally in Modern English?

Over the course of a few hundred years, various different changes of different natures conspired to produce this perfectly symmetric system with voicing distinctions at all places.

1. Initially, borrowings from French and from other Old English dialects that allowed initial voiced fricatives led to words like "vat" and "vixen" entering the language, creating a contrast word-initially with words like "fat" and "fox".

2. Medially, geminates (double-consonants) which didn't undergo the original voicing rule became simplified. So, cyssan "kiss" stopped having a "long" voiceless pronunciation and instead shortened to our modern "short" voiceless pronunciation, putting it again in contrast with the voiced "z" medially that was the result of the initial voicing rule.

3. Finally, final vowels slowly reduced and then dropped off, as in words like bathe vs. bath, leaving voiced consonants in contrast with voiceless ones word-finally.

So, the change started as a perfectly symmetric system of voiceless consonants in all positions, initial, medial, final:

F : F : F

Changed to a predictable (allomorphic) distribution due to regular medial voicing:

F : V : F

And then ultimately developed into another perfectly symmetric (phonemically distinct) system due to the conspiracy of various different borrowing and sound changes processes:

F/V : F/V : F/V

(Btw, these "conspiracies" are one of the advantages of Optimality Theory over older Rule-Based Phonology, which I can discuss more if anyone's interested).

So, yeah, regardless of the orthography (which varied both between writers and over the course of the period), thorn was the sound initially and finally, and eth was often the sound medially in Old English.

Initially eth was the voiced dental fricative (as in "them") and thorn was unvoiced (as in "thin")

Two different sounds, þ is th like in thought, while ð is th like in the.

Þis is þe rule in Icelandic.

Why is it that when reading this sentence, any word that has the thorn in it that could start with p was autocorrected in my brain, but the words that obviously begin with th were not?