There is a bias against the genitive case with inanimate things, that is sometimes found in advice to avoid it in some cases. In some cases that advice is indeed, that one should only use it with people and sometimes that one should only use it with living things. (So "the dog's" is allowed, but "the car's" is not).
Fowler raged against it, and blamed headlines' need for brevity (or as he would rather say, the need for brevity of headlines).
The rule was never consistent, for some inanimate words of one syllable would generally be accepted as being allowed with such a construct, but there really wasn't a clear rule expressing which these were.
This has largely died away, with those who favour the apostrophe form winning, but it does linger and some people will still give the advice that you "seem to remember".
This disagreement stems from an earlier disagreement in early Modern English, that in turn stems from one in Middle English.
In Middle English, his served as both neuter and masculine genitive pronoun. During this period, some used it (note the lack of any s) for the neuter genitive pronoun, but in formal use, it remained his into the Modern English period.
Now, in the beginning of the Modern English period, some started using it's that applies the genitive with an 's to it much as Fred's does to Fred (indeed while it's now considered a classic error to use it's when you mean its, this didn't die out entirely until the 1800s), some used its - dropping the apostrophe and gaining distinction from it's meaning it is along with mirroring theirs, hers etc., and some where still using his.
So, much as we today may wonder whether we should use e-mail or email, C.D.'s or CDs and whether it was okay to start a sentence with However, because there is a difference of opinions, so too would somebody writing between 1604 and 1611 writing a work that they wanted to be in plain but respectable English wonder whether they should use his or its.
On the one side there would be those who said that its was now more common, and more clearly understood as having a neuter gender while his was rather stuffy and not the vernacular speech of the common man (an important point to 17th Century Protestants). On the other would be those who said that his is the formal form, and hence more respectable, and besides which if they wanted this work to still read clearly in decades to come then they should avoid slangy new words like its that might die out again and leave people saying "dude, the 1610s called, they want their bible back". (This explanation may have more words and expressions that made no sense in the early 17th Century than it strictly should).
For this reason, the King James Bible avoids use of both its and his (in the neuter sense) and instead favoured a very heavy use of of and thereof.
So, where the New Revised Standard Version has Genesis 2:19 as:
So out of the ground the Lord God formed every animal of the field and every bird of the air, and brought them to the man to see what he would call them; and whatever the man called every living creature, that was its name. [Emphasis mine]
The King James Version has it as:
And out of the ground the Lord God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof. [Emphasis mine].
Now, the KJV did not avoid the genitive apostrophe itself:
And he looked, and behold a well in the field, and, lo, there were three flocks of sheep lying by it; for out of that well they watered the flocks: and a great stone was upon the well's mouth.
However, clearly avoiding the genitive when you would be using a neuter pronoun will reduce the number of times you use the genitive with a non-living thing (and with a living thing of indeterminate gender), and increase the number you favour other constructs. And the King James Version bible has had a long history as being the model for what people understand good English and/or formal English to sound like. It was the only book many people read or heard read, and even for the educated it was the single book that many read or heard read most often.
Hence people became used to hearing the genitive avoided in many cases of non-living things or animals of indeterminate gender. Hence even when the pronouns weren't used, people came to avoid the genitive in such cases.
So even when the genitive would not involve a pronoun at all, people would favour other constructs. This also goes some way to explain the love lawyers have of horribly convoluted sentences that end with thereof - while it does add precision in some cases, in many it's just a form of hyper-formal use that is trying to sound clever (this isn't necessarily an insult to lawyers, in many cases when you engage a lawyer you don't just want them to make their case for you, but also to convince the other party that you have a clever lawyer, so that's actually part of their job).
Ironically though, this history also argues against it. The difficulty someone in the 16th through to the 17th Centuries had in picking between two neuter genitive pronouns in itself shows that it was permissible to use genitive inflection with non-living things, and always has been. If we can do it with pronouns (and we now have its quite clearly as the choice to go for), then we can do it with nouns. And there's the fact that the KJV used it itself (h/t to Peter Shor).
So, in all, "The car's antenna" is fine, "The antenna of the car" would still be favoured by some, and "Its antenna" is also fine, though the KJV would have used "the antenna thereof".