The short answer to the question is - the ESV accurately translates the operative word מִדְבָּר See appendix below for BDB meaning. This word occurs about 271 times in the OT and is mostly translated "wilderness", eg, Gen 14:6, 16:7, 21:14, 21:20, etc.
The question SHOULD be, Why does the KJV translate the word as "south". I do not know the answer to that question. However, the desert/wilderness closest the Jerusalem is south from there and thus, the KJV translators used an unnecessary interpretation and substituted "south".
APPENDIX BDB meaning of מִדְבָּר
I. [מִדְבָּר] noun masculine mouth, as organ of speech, — מִדְבָּרֵךְ נָאוֶה thy mouth is lovely Songs 4:3 "" שִׁפְתוֺתַיִךְ, ᵐ5 λαλιά, Jerome eloquium.
II. מִדְבָּר270 noun masculine wilderness — Deuteronomy 32:10 +; with ה locative מִדְבָּ֫רָה Joshua 18:12 15t.; construct Exodus 15:22 +; מִדְבַּ֫רָה 1 Kings 19:15; suffix מִדְבָּרָהּ Isaiah 51:3; —
1 tracts of land, used for the pasturage of flocks and herds, דָּֽשְׁאוּ נְאוֺת מִדְבָּר the pastures of the wilderness put forth green grass Joel 2:22; ׳מ ׳יִרְעֲפוּ נ the pastures of the wilderness drop (fertility) Psalm 65:13; ׳מ ׳יָָֽבְשׁוּ נ are dried up Jeremiah 23:10, compare Jeremiah 9:9; Joel 1:19,20.
2 uninhabited land, מִדְבָּר לֹאאָֿדָם בּוֺ wilderness in which is no man Job 38:26; the abode of pelicans Psalm 102:7; wild asses Job 24:5; Jeremiah 2:24; jackals Malachi 1:3; ostriches Lamentations 4:3; מִייִֿתְּנֵנִי בַמִּדְבָּר מְלוֺן אֹרְחִים וְֶ˜אעֶזְבָה אֶתעַֿמִּי O that I had in the wilderness a lodging place of wayfarers, that I might leave my people Jeremiah 9:1; טוֺב שֶׁבֶת בְּאֶרֶץ מִדְבָּר better to dwell in a desert land, than with a contentious woman Proverbs 21:19; בְּאֶרֶץ מִדְבָּר וּבְתֹהוּ יְלֵל יְשִׁמֹן in a desert land, and in a waste howling wilderness Deuteronomy 32:10.
3 large tracts of such land bearing various names, in certain districts of which there might be towns and cities: יִשְׂאוּ מִדְבָּר וְעָרָיו חֲצֵרִים תֵּשֵׁב קֵדָר let the wilderness and its cities lift up (their voice), the villages that Kedar doth inhabit Isaiah 42:11. There were six cities in the wilderness of Judah Joshua 15:61,62; הַמִּדְבָּר usually = wilderness of the wanderings Genesis 14:6; Numbers 14:16,29,32,33 (twice in verse) +, or the great Arabian desert Judges 11:22 +; but may also refer to any other Songs 3:6; Songs 8:5. Special tracts — (a) of the wilderness of the wandering were שׁוּר ׳מ Exodus 15:22, סיןExodus 16:1 +, סיני Exodus 19:1 +, פארן Numbers 13:26 +, צן Numbers 20:1 +, קדשׁ Psalm 29:8, אתם Numbers 33:8; (b) in West Palestine יהודה ׳מ Judges 1:16; Psalm 63:1 compare Joshua 15:61, מעוֺן 1 Samuel 23:24,25, זיף 1 Samuel 23:15; 1 Samuel 26:2, באר שׁבע Genesis 21:14, עין גדי 1 Samuel 24:2, תקוע2Chronicles 20:20, ירואל2Chronicles 20:16, גבעון 2 Samuel 2:24; (c) in Eastern Palestine מואב Deuteronomy 2:8, אדום 2 Kings 3:8, קדמות Deuteronomy 2:26.
4 figurative וְשַׂמְתִּיהָ כַמִּדְבָּר וְשַׁתִּהָ כְּאֶרֶץ צִיָּה and (lest I) make her as a wilderness and set her like a dry land Hosea 2:5; הֲמִדְבָּר הָיִיתִי לְיִשְׂרָאֵל have I been a wilderness to Israel ? Jeremiah 2:31.
דבשׁ (√ of following; compare Arabic IX. become black or brown, of a colour between black and red; see derivatives; so Fl see below; yet verb in this meaning perhaps denominative from i.e. having the colour of dibs).