How important are Perf rating and Raider IO Score for getting into pugging HC/Mythic/M+ content
Solution 1:
From my experience in this and past expansions most leaders are too lazy to check for scores. If you have the proper achievements you get invited in most cases. This might be different in mythic raid pugs.. I haven't tried those yet.
If you have the Curve achievement (Ahead of the Curve: [latest final boss here]; kill the final boss on heroic difficulty) you can pretty much join every pug raid in NM and HM. Logs are mostly irrelevant and only a very few raid leaders check for those. When I posted the Cutting Edge achievement (kill the final boss on mythic difficulty) it was usually an instant invite when the gear was ok... There wasn't enough delay to assume they checked my logs.
In M+ it's mostly about the character and class. Some classes get no invite at all, regardless of their performance logs and gear. With the right class they still want high gear and the achievement Keystone Master (complete a +15 in time) and after that it's pretty easy.
In short: Good achievements and gear are worth more than performance logs.