Arguably, the KJV is a modern Christian text.

At any rate, this isn't some great conspiracy by translators to "expunge" mythical animals in the Bible as you're implying. Rather, as modern scholarship gets better insight into the original meaning of the Greek and Hebrew, translators update later translations of the Bible. Archaeologists and Biblical scholars are constantly making new discoveries, and reassessing previous interpretations as more work is done on the text (as various scholars do in a wide range of ancient texts, not just Biblical).

So, in your unicorn example, the Latin Vulgate used the word unicorn or rhinoceros as a translation for the Greek word monokeros (one-horned) found in the Greek version of the Old Testament called the Septuagint (aka LXX).

But the Septuagint translated monokeros from the original Hebrew word reem ‪(רְאֵם), which we now know was a type of extinct ox-like animal called an aurochs.

In his commentary on Job, Biblical scholar David J. A. Clines has this to say about the creature,

The wild ox (רְאֵם) was understood as the unicorn by the LXX (so too KJV), and the rhinoceros by Aquila and the Vulgate, but is now generally identified with the aurochs (Bos primigenius), the ancestor of most of our modern cattle, but now extinct (the last specimen is said to have died in 1627). This massive animal was about 10 feet long, and could weigh over a ton, half the size of a rhinoceros; it had two thick, long, curved horns. Like all oxen, the aurochs was herbivorous, and frequented woods and hills; it was usually found in small herds, though males were sometimes solitary.

The aurochs was hunted in Egypt, Palestine, and Babylonia. It is usually identified with the rimu mentioned often in the annals of the Assyrian kings. On the Ishtar Gate at Babylon (ANEP, fig. 760), the blue tile decoration is of alternating rows of aurochs and dragons. In Palestine, Baal hunts aurochs in the marshes of Lake Huleh (Baal I.i19 [CML, 109b]; III.vi.18 [CML, 115a]; Aqhat II.vi.20 [CML, 55al]). In classical literature, see Herodotus, Hist. 7.123 (in Macedonia); Aristotle, Hist. anim. 1.1 (488a); 2.1 (499a); Pliny, Natural History 8.30.72 (in Ethiopia).

In the Hebrew Bible the huge and dangerous horns of the aurochs are especially mentioned (Num 23:22; 24:8; Deut 33:17; Pss 22:21 [22]; 92:10 [11]); cf. Also Pss 29:6; 34:7. In Deut 33:17 the horns of a domesticated ox are praised as being like those of an aurochs. Here, since it is a wild beast, not only is the aurochs untameable (vv 9-10), but it is unreliable also (vv 11-12).

Bibliography,

Buren, E. Douglas van. The Fauna of Ancient Mesopotamia as Represented in Art. AnOr 18. Rome: Pontificium Institutum Biblicum, 1939. 69-74. Cansdale, George. Animals of Bible Lands. Exeter: Paternoster Press, 1970. 82-84. Godbey, Allen H. "The Unicorn in the Old Testament." AJSL 56 (1939) 256-96. Hilzheimer, Max. Die Wildrinder im alten Mesopotamien. Mitteilungen der Altorientalischen Gesellschaft 2/2. Leipzig: E. Pfeiffer, 1926. Keller, Otto. Die antike Tierwelt. Leipzig: Engelmann, 1909-13. 1:341-43. Klotz, John. "Notes on the Unicorn." CTM32 (1961) 286-87. Ley, Willy. The Lungfish, the Dodo and the Unicorn. New York: Viking, 1948. 19-34. Muller, H.-P. "רְאֵם." THWAT, 7 (1990) 267-71 [= TDOT, 13 (2004) 243-47]. Velden, Adria Schouten van der, and Adrian Schouten van de Velden. Tierwelt der Bibel. Stuttgart: Deutsche Bibelgesellschaft, 1992. 24-25.

I can get into some of the other creatures when I have more time. Of course, not all the names of all creatures (mythical or otherwise) have been reassessed. Some have, some have not. It really depends on where academic work on the subject takes scholars.

You should post this question on /r/AcademicBiblical

Leviticus 11:13 is a pretty obvious translation error to render it in English as "griffon".

The Torah Hebrew for this verse (with masoretic vowels, cantilation, etc) is:

וְאֶת־אֵ֨לֶּה֙ תְּשַׁקְּצ֣וּ מִן־הָע֔וֹף לֹ֥א יֵאָֽכְל֖וּ שֶׁ֣קֶץ
הֵ֑ם אֶת־הַנֶּ֨שֶׁר֙ וְאֶת־הַפֶּ֔רֶס וְאֵ֖ת הָֽעָזְנִיָּֽה:

The word here being translated as "griffon" is פרס, which in modern Hebrew refers to the species Gypaetus barbatus, known in English as a bearded vulture or ossifrage.

Regardless of if modern Hebrew preserve the exact species that was meant by the Torah (who knows?), it's pretty clear from the context that this word is intended to refer to a real-world bird of prey, probably a scavenger of some sort, with which the audience would have at least a passing familiarity, which is why the verse is instructing you not to eat them because they are considered unclean.

How did פרס get translated as "griffon"? I don't know exactly how, but I blame Jerome on basic principles.