Just stating the facts.
Chess.com mute policy for private group admins

I had a similar problem in March/April. I was muted for 3 days after making an announcement in the group and inviting people to the Annual Daily Tournament.
I did not send a private message to the members, I just used the option that Chesscom itself provides to send a message to members who wish to receive this type of communication. And I didn't post a lot (about 1 ad a month and 1 new forum a week).
Many members wanted to talk to me about tournaments or other relevant things and I was muted.
And, what's worse: when we are muted, all posts (ads, forum threads, blogs,...) are deleted or hidden. Then, all club posting material (relevant material about chess and tournaments) is hidden and all club members are at a loss for what to do.
I spent time contacting Support trying to resolve this issue generated by Chesscom itself. So they solved it in 35 hours (instead of 3 days). But now I feel unsure if I do post on the Forum, Announcements, New Tournaments (I have no idea if I do or not).

It would the equivalent members being able to block chess.com staff and moderators, such that staff would be unable to send warning messages and such. Club Admins basically function as the staff members of their own microcosms of chess.com.
Good one LOL but I'm sorry to inform you staff and Mods can not be blocked in any way But blocking us would not solve this problem and matter of fact if any admin get muted for sending club messages Please Direct Message me with the link. That should not be happening if not true spamming and gerally admin with paid accounts were given a higher limit for that reason as admins do have to send the same message to just selected members. Thanks

It would the equivalent members being able to block chess.com staff and moderators, such that staff would be unable to send warning messages and such. Club Admins basically function as the staff members of their own microcosms of chess.com.
Good one LOL but I'm sorry to inform you staff and Mods can not be blocked in any way But blocking us would not solve this problem
This was exactly my point XD Since members can't block chess.com staff from sending messages, they shouldn't be able to block club Admins from sending messages, either.

....matter of fact if any admin get muted for sending club messages Please Direct Message me with the link. That should not be happening if not true spamming and gerally admin with paid accounts were given a higher limit for that reason as admins do have to send the same message to just selected members. Thanks
If the same message is sent to a lot of members in relatively quick succession, it absolutely will mute the account via the auto-mod.
I have been muted for it before.

This was exactly my point XD Since members can't block chess.com staff from sending messages, they shouldn't be able to block club Admins from sending messages, either.
My guess is that won't change either. If members are getting messages when they have their settings set to not receive them, that would also cause support requests and bug reports.

This was exactly my point XD Since members can't block chess.com staff from sending messages, they shouldn't be able to block club Admins from sending messages, either.
My guess is that won't change either. If members are getting messages when they have their settings set to not receive them, that would also cause support requests and bug reports.
Indeed Martin_Stahl is right about the speed of the messages. Years ago before my Mod days I was Admin in 75 clubs was running me crazy but I do remember now that we calculated it one time that if you waited 15 secs between messages you wouldn't be nabbed by the spam bot as far as I know that hasn't changed. If you are sending many messages do it slowly and avoid the spam issue.

People ban people since they power hungry
The only people I ban from my clubs are those who refuse to cooperate.

I support this proposal as the OP states.
Does anyone here know for certain if emails sent per message by default? IMO it shouldn't be if it is. Chess.com has anti-spam laws to abide and domain rep scores to uphold, lest chess.com get blacklisted for spam-like behaviour. That said, admins using messaging to administer a club is clearly not unsolicited email regardless. However, tweaking default settings to only send emails for those who opt-in, would greatly reduce any false positive risks for chess.com and their email domain reputation.

Once I had a crucial live match for my team, but got muted for sending the link of the match to a team member a few days before, and couldn't contact anyone in the team for a week or so till the mute was over, and hence no one joined the match because I couldn't remind them a few hours before the match....So I agree 100% with what @tricky_dicky says, the members should just leave the club if they don't want to get notified by us. Chess.com must not interfere.

If you don't want people joining your club and then just going inactive/dead, make it joinable only on request and ask people if they will be active before joining.
There is nothing else you can do on that issue.
Some people just want to be in clubs just for fun and to just check out stuff and not to play matches.

Chesscom has a option where the members can mute the notifications. On Profile Settings and on Club Settings too. Then, Chesscom seems very contradictory.

As it's been said in this thread, probably the notifications by email are the most annoying and should be disabled by default.
Bottomline IMHO the whole notification system, website-wide, should be rethought.

@Martin_Stahl
The announcement tool is a good option but it acts like a big hammer hitting all members (except those with it switched off) and sometimes Admins need to be more selective and focused on individual members.
You also stated :- ‘it would seem the site is most interested in being sure members are allowed to limit the messages they receive and I can't imagine a blanket allowance for club admins to send all members messages’
This is an illogical position. In this group you are moderating, and it would be ridiculous for you to be muted. That is the same position for Admins in a private group. C.C have stated many times they will not interfere with private group but then have a policy which restricts said Admins for performing that task on their behalf. If a member doesn't want information from the group admins then why are they in the group? They can always leave.
(For clarification I am not advocating immunity for Admins from muting. If they are spamming site members generally then the mute still needs to be applied. Own group messages should be exempted from spam points)
@Master_Merlin
You stated :- ‘if any admin get muted for sending club messages Please Direct Message me with the link. ‘
How can we message you if we are muted? I believe that when muted you cannot message any other member, Mods and Staff included.

GM DJ_Haubi has shared his thoughts on why he is no longer supporting Chess.com groups.
https://www.chess.com/clubs/forum/view/whats-our-situation-whats-to-do-next?page=2#comment-88580085

It would the equivalent members being able to block chess.com staff and moderators, such that staff would be unable to send warning messages and such. Club Admins basically function as the staff members of their own microcosms of chess.com.
Good one LOL but I'm sorry to inform you staff and Mods can not be blocked in any way But blocking us would not solve this problem and matter of fact if any admin get muted for sending club messages Please Direct Message me with the link. That should not be happening if not true spamming and gerally admin with paid accounts were given a higher limit for that reason as admins do have to send the same message to just selected members. Thanks
That's hilarious. How is a muted member supposed to direct message you? 🤔

Actually a very good question. I guess you need to understand how the site mute function works. If your muted in a private club you are only muted for that club and it doesn't interfere with your ability to post anywhere on the site except for that club. It takes a MOD or Staff member to mute you site wide a club admin doesn't have all the Mod tools the Mods and staff have. So if you have been muted by a club no big deal on the site. On the other hand if muted by a Mod or staff then you have to contact support for help by email at support@chess.com. Knowing all the info makes my statement a bit more understandable..
Thanks for that, Arjun1516. Really good contribution to the subject matter.