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#JavaScript (ES6), 203

Implementing this algorithm (point 3, totally iterative)

Within the limits of script executing in Firefox, I managed to have the 200K console output for input 12 in about 1.5 minutes

n=>{
for(b=[...Array(n)].map((_,i)=>1+2*i),z=0;!z;)
for(c=[...'('.repeat(n+n)],b.map(n=>c[n]=')'),console.log(c.join``),z=i=n;i--&&z;)
if(b[i]<n+i)for(z=0,b[j=i]++;++j<n-1;)b[j]=Math.max(b[j-1],2*j)+1
}

Less golfed

n=>{
  b=[...Array(n)].map((_,i)=>1+2*i)
  q=0
  for(z=0;!z;)
  {
    c=[...'('.repeat(n+n)],
    b.map(n=>c[n]=')'),
    console.log(++q,c.join``)
    for(z=i=n;i--&&z;)
      if(b[i]<n+i)
        for(z=0,b[j=i]++;++j<n-1;)
          b[j]=Math.max(b[j-1],2*j)+1
  }
}

Test generating all output but do not display it, just counting output rows

I got 90 sec for input 15

F=n=>{
for(b=[...Array(n)].map((_,i)=>1+2*i),z=0;!z;)
for(c=[...'('.repeat(n+n)],b.map(n=>c[n]=')'),console.log(c.join``),z=i=n;i--&&z;)
if(b[i]<n+i)for(z=0,b[j=i]++;++j<n-1;)b[j]=Math.max(b[j-1],2*j)+1
}

console.log=x=>o++ // console.log display no output, just count rows

function test() {
  n=+I.value,o=0,t=+new Date
  O.textContent='...wait...'
  setTimeout(_=>(F(n),O.textContent=['rows='+o,'time sec '+(new Date-t)/1000]), 10)
  
}

test()

N:<input id=I value=12 type=number max=20><button onclick='test()'>go</button>
<pre id=O></pre>