First: The present tense in Hebrew does NOT necessarily signal present time. It signals durative or continuous action.
Second: The present tense is in fact NOT used. What is used is the Present Active Participle, which signifies action which continued up to a point of time and stopped.
וְנָהָר֙ יֹצֵ֣א מֵעֵ֔דֶן should be read: And a river used to flow from Eden.
I do not know Hebrew, but only ESV and one paraphrase use the past tense as you did:
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Septuagint:
And the name of the second river is Geon, this it is which
encircles the whole land of Ethiopia.
New International Version
The name of the second river is the Gihon; it
winds through the entire land of Cush.
New Living Translation
The second branch, called the Gihon, flowed around the entire land of Cush.
English Standard Version
The name of the second river is the Gihon. It is the one that flowed around the whole land of Cush.
New American Standard Bible
The name of the second river is Gihon; it
flows around the whole land of Cush.
King James Bible
And the name of the second river is Gihon: the same is it that
compasseth the whole land of Ethiopia.
Holman Christian Standard Bible
The name of the second river is Gihon, which
flows through the entire land of Cush.
International Standard Version
The name of the second river is Gihon— it
winds through the entire land of Cush.
NET Bible
The name of the second river is Gihon; it
runs through the entire land of Cush.
New Heart English Bible
The name of the second river is Gihon; it
goes around the whole land of Cush.
GOD'S WORD® Translation
The name of the second river is Gihon. This is the one that
winds throughout Sudan.
JPS Tanakh 1917
And the name of the second river is Gihon; the same is it that
compasseth the whole land of Cush.
New American Standard 1977
And the name of the second river is Gihon; it
flows around the whole land of Cush.
Jubilee Bible 2000
And the name of the second river is Gihon; this is the same that
compasses the whole land of Ethiopia.
King James 2000 Bible
And the name of the second river is Gihon: the same is it that
encompasses the whole land of Cush.
American King James Version
And the name of the second river is Gihon: the same is it that
compasses the whole land of Ethiopia.
American Standard Version
And the name of the second river is Gihon: the same is it that
compasseth the whole land of Cush.
Douay-Rheims Bible
And the name of the second river is Gehon: the same is it that
compasseth all the land of Ethiopia
Darby Bible Translation
And the name of the second river is Gihon: that is it which
surrounds the whole land of Cush.
English Revised Version
And the name of the second river is Gihon: the same is it that
compasseth the whole land of Cush.
Webster's Bible Translation
And the name of the second river is Gihon: the same that
compasseth the whole land of Cush.
World English Bible
The name of the second river is Gihon: the same river that
flows through the whole land of Cush.
Young's Literal Translation
and the name of the second river is Gibon, it is that which
is surrounding the whole land of Cush;