I think the answer lies in a conceptual problem: We, as moderns, fundamentally conceive our words as written. To paraphrase Walter Ong in his Orality and Literacy I challenge you to imagine a word without defaulting to an image of it on a page. It's almost impossible. We learn words most often by reading them, looking them up in a dictionary, and then learning to pronounce them. The usual word learning process is, therefore, this: Step 1. Learn the "form" of the word, its spelling and its existence, Step 2. Learn the meaning of the word, Step 3. Learn the pronunciation. What this means is that for most moderns, the fundamental nature of a word is its written form and its meaning, whereas pronunciation is somewhat incidental. It is perfectly possible, and, at least for me, common, to know words without knowing how to say them, because you've only read them and never looked up the pronunciation.

Medieval Europe, however, was not this way. Fundamentally, Medieval Europe was a primarily oral culture. Most people, and in some times and places even most elites could not read or write. Even when writing was possible, speech was seen as both more fundamental and more powerful: it was common practice to send someone a formulaic and somewhat unimportant letter, but send an oral message along with it that carried the real meat of the interaction. It's hard to wrap your head around this, because we're so wedded to text, but for people like that, the sound of a word is its fundamental form. If all you have is speech, than the spoken word is the form of language.

Now, Early Modern Europe is a difficult case, particularly in the 15th and 16th centuries. Why? Because, as you note, it's NOT an exclusively, or even necessarily a primarily oral culture anymore, but it is one in transition. Printing and paper have driven down the cost of books tremendously, just as the state was becoming denser and demanded more administrative personnel, meanwhile, cities, towns and trades are growing rapidly, and education is improving, all rapidly pushing up the literacy rate. What this all means is that while society is becoming much more literate, the change over isn't fully complete yet, and this is where we get to spelling.

Vernacular English, when written, had long been spelled using the Latin alphabet as a set of phonemes to express the fundamentally spoken language. People wrote things how they sounded, and authors frequently spelled the same word differently within their own work. There was no standard whatsoever, and no one expected there to be. There never had been. No one saw a problem with this because of that oral orientation I wrote of earlier in this answer: if words were fundamentally spoken, it didn't really matter much if you spelled things differently, just as we today don't really struggle with people pronouncing the same word differently, just so long as it's spelled correctly. All you were doing with spelling was expressing the sounds of the spoken word.

As a brief digression, this is also why much English spelling is unphonetic today. Simultaneous with the standardization slowly taking hold throughout the 16th century, English was still the process of a major pronunciation change known as the Great Vowel Shift. For reasons more or less unknown, almost every vowel in English substantially changed in pronunciation between 1300 and 1700, along with numerous other more minor changes. However, this change was still in progress when spelling finally was mostly ossified, which means that spelling now reflected old pronunciations. The best example I can think of off the top of my head is the word Knight, which was originally pronounced kuh-nicht, with the ich being pronounced similarly to the word in German today. The first syllable became silent, the vowel in the second changed, but the spelling remained.

This all brings us to names: people wouldn't necessarily feel that they'd gotten their "true name" wrong, because their "true name" was the SOUND, not the written word. If that's true, then a man named Joseph would have no reason to mind Josef, Joseph, Joesef, or Joesef, or whatever other spelling someone concocted. It expressed the sound correctly, which meant it was correct. I believe the 16th century is the tail end of this trend among the educated and literate, though I noticed this sort of phonetic spelling in civil war letters I encountered in a book, suggesting it persisted in less educated writers for a long time.

The last note I'd make is that it's easy, because of our cultural predispositions, to see this as strange. We live in an overwhelmingly literate age, and it's fairly easy for me to speak to a friend or spend a day communicating mostly via written words, but this is fundamentally pretty new, and arguably not the natural thing. Certainly, children, when they're learning to spell, spell mostly by sounding words out, and older folks can often be heard complaining about how nonsense English spelling really is. If we didn't think that the written word was fundamental, and spoken incidental, than that might not be the case.