Output the nth 'boring' number
JavaScript (ES6),  226 220  209 bytes
Saved 11 bytes thanks to @HermanL
Input is 1-indexed.

f=(x,n=m=c=1,D=k=&gt;k?n%k?D(k-1):[k,...D(k-1)]:[],S=a=&gt;s=eval(a.join`+`))=&gt;(S(a=D(n))&gt;m?m=s:0)|(s=a.length)&lt;3|(s&lt;c?0:c=s)|a.reduce((a,x,i)=&gt;i?[...a,...a.map(y=&gt;[...y,x])]:a,[[]]).some(a=&gt;S(a)&gt;=n)||--x?f(x,n+1):n

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Optimizations are based on the following properties:

All primitive abundant numbers are either semiperfect or weird.
All highly composite numbers are largely composite.
All superabundant numbers are highly abundant.