C (gcc), ^{117 \$\cdots\$ 111} 101 bytes

Saved 10 bytes thanks to an idea from Davide!!!

i;f(char*p){for(i=0;*p++-40||p[3]-41;++i);p[-1]=p[1]-62?*p:p[2];memcpy(p,p+4,strlen(p));i&&f(p-i-1);}

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C (gcc), ^{117 \$\cdots\$ 101} 98 bytes

Saved 10 bytes thanks to an idea from Davide!!!
Saved 3 bytes (and got below 100!) thanks to ceilingcat!!!

i;f(char*p){for(i=0;*p++-40||p[3]-41;++i);p[-1]=p[p[1]-62?0:2];bcopy(p+4,p,strlen(p));i&&f(p+~i);}

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C (gcc), ^{117 113} 111 bytes

i;f(char*p){for(i=0;p[i++]-40||p[i+3]-41;);p[i-1]=p[i+1]-62?p[i]:p[i+2];memcpy(p+i,p+i+4,strlen(p));~-i&&f(p);}

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Explanation

C (gcc), ^{117 \$\cdots\$ 111} 101 bytes

Saved 10 bytes thanks to an idea from Davide!!!

i;f(char*p){for(i=0;*p++-40||p[3]-41;++i);p[-1]=p[1]-62?*p:p[2];memcpy(p,p+4,strlen(p));i&&f(p-i-1);}

Try it online!

Explanation (before some golfs)

Takes an input string with enclosing in parentheses and reduces the input string to either a left (<) or a right (>).

Takes an input string with enclosing parentheses and reduces it to either a left (<) or a right (>).