I know one of the PDX folks have commented on a steam, that while open betas are great for finding bugs and broken things, they are a nightmare to manage, because 90% of the things, that get reported are duplicates, or the quality of the report is low, lackluster, doesn't even state steps to duplicate.
If what PDX said was true, then I don't know how they manage to comb through their live bug reports. There is a greater order of magnitude more bug reports on a live game than a beta one. With regards to reporting bugs, PDX puts many barriers (inconveniences) in place which makes it a hassle. There's no in-game bug reporting button, so you have to tab out and write up a report on their website. (Which you have to log into) Furthermore, it requires information like version#, checksum, and in some cases the player needs to find where their screenshots and saved games are kept.
These don't seem like much of an inconvenience, but the chances are that a majority of the people who might report a bug will give up upon immediately learning there is no bug reporting button and that they have to go to a website to do it. Having a beta also culls out a large amount of people, especially if you have to opt into it.
What you're describing is basically a Friends & Family alpha, which PDX probably already has. I have participated in a few of those and one of the biggest problems that you run into is leaks and security. Close friends and families of developers can be trusted, but can you trust some "high profile" member of the community to not share what's happening to his friends secretly on his clan's Discord? I've seen many futile measures take place to tighten up leaks and at the end of the day it's almost impossible to keep strangers from sharing secrets with their friends.
A bigger problem (imo) is the fact that it's really exploitative, taking advantage of the passion of gamers. Paradoxically (ha!), the more professional and skilled a group of volunteers gets at reporting and diagnosing bugs, the more ethically questionable it becomes. A secret group of volunteer testers is a thankless, perkless job that will often be demoralized by the bugs that inevitably slip through which the community will lambast about.
In my opinion, the fairest way to do a beta is just to have it be NDA free and open. (Or at the worst, opt-in) Good bug reports will rise to the top, and if the program goes on long enough it becomes easier over time for developers to recognize members of the community that are proficient at reporting things.