Wright has already attempted this: He has a fork of Bitcoin called "Bitcoin SV" (satoshi's vision) which he claims is the real Bitcoin and that Bitcoin is an imposter. Today it has well under 1% of Bitcoin's hashpower, well under 1$ of Bitcoin's price (and has fallen a LOT relative to Bitcoin), and it's now down to less than 40 reachable nodes. Most of the few exchanges that have ever had it have dropped it, and it has recently had suffered some extremely severe reorg attacks -- causing millions of dollars in losses to at least one exchange.
For the purpose of Wright's backdoor demands miners are irrelevant. If contributors were all anonymous you'd have the problem that they might actually be wright. He's even suing former developers like me, who haven't contributed for a couple years (or more!) ... really the targets of the lawsuit don't have much connection to reality. And the more obscure you make it to get the software the more ability he'd have to just setup impersonation sites to get people to run his malicious version instead.
Also, anonymity hasn't helped that much in the case of the owner of Bitcoin.org-- being anonymous just caused him to lose in a default judgement when Wright sued him because the UK will not allow an anonymous party to defend themselves. Ultimately they'll likely have to deanonymize to prevent Wright from seizing Bitcoin.org.
Addresses being private wouldn't really change anything about Wright's attack. He'd still demand to be given coins just the same. In fact, as of the moment Bitcoin is anonymous enough to severely frustrate his efforts: He claims to "own" 1.1M BTC but right now he's only suing developers for a backdoor to acces 0.111M of it, because whenever he'd claimed specific addresses for more he's gotten nailed by the true owners signing messages. So right now he's only going after a smaller portion where he can be confident the current controlling party won't speak up (the bulk of those coins were stolen from MTGox and haven't moved since).