JavaScript (Node.js), 55 bytes

f=(a,b=a.replace(/\((.>)?(.)(<.)?\)/,'$2'))=>a<b?f(b):b

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Repeat apply regex replace until no match found. The regex /\((.>)?(.)(<.)?\)/ get the calculate result as 2nd capturing group.

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JavaScript (Node.js), 50 bytes

f=a=>a[1]?f(a.replace(/\((.>)?(.)(<.)?\)/,'$2')):a

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Repeat apply regex replace until only 1 char left. The regex /\((.>)?(.)(<.)?\)/ get the calculate result as 2nd capturing group.

Save 1 byte, thanks to Arnauld

JavaScript (Node.js), 55 bytes

f=(a,b=a.replace(/\((.>)?(.)(<.)?\)/,'$2'))=>a<b?f(b):b

Try it online!

Save 1 byte, thanks to Arnauld

JavaScript (Node.js), 55 bytes

f=(a,b=a.replace(/\((.>)?(.)(<.)?\)/,'$2'))=>a<b?f(b):b

Try it online!

Repeat apply regex replace until no match found. The regex /\((.>)?(.)(<.)?\)/ get the calculate result as 2nd capturing group.

Save 1 byte, thanks to Arnauld

JavaScript (Node.js), 60 bytes

f=(a,b=a.replace(/\(...\)/,c=>c[c[2]>'='?3:1]))=>a==b?b:f(b)

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JavaScript (Node.js), 55 bytes

f=(a,b=a.replace(/\((.>)?(.)(<.)?\)/,'$2'))=>a<b?f(b):b

Try it online!

Save 1 byte, thanks to Arnauld