I am not in a campaign yet but I guess I know why it's a problem if everyone is getting paid to the same address for as long as possible. That's basically a privacy leak. But we can not tell whether some of the participants make use of silent payments or managers using LN for pay out to contain this.
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It doesn't really matter that every payout from a campaign commonly is sent to one address per participant. In the application process applicants publish their receiving address here in the forum. So it's basically public and attributable to a forum account.
Even if it were non-public, e.g. via PM, the public weekly results, spreadsheets and payment transactions would make it rather easy to attribute receiving addresses to campaign participants. So it would make not much sense to try to keep it "secret". It would be error-prone more work for basically little to no obscurity.
Later if and when you consolidate the payments, you could use coinjoins or other methods to obfuscate the origin of those coins. It depends on what you want to achieve or what you want to hide.
All on-chain Bitcoin transactions are public. The question is, are they also attributable to individual entities?
Regarding Electrum servers:
For various reasons (and because I can) and privacy is one of them, I operate my own full non-pruned node and Electrum server and local instances of blockchain explorers like mempool.space and bitcoinexplorer.org likes. Whatever I do with my wallet's addresses or with some blockchain digging, it stays local and nobody else can log stuff.
