Silent Payments only hide the full balance. But people will still be able to collect all silent payments with a zap event and merge that into your "balance" in a similar way they do it with lightning right now.

Is that so? People see you received cause it was a zap but nobody sees if you sent the funds out again.

they can just check the chain for outputs.

Right, so public zaps can be observed and added up to estimate someone's public balance, and this shows us that there are different concepts that we're trying to think about but they are not being clearly distinguished so people end up talking past each other sometimes. We can distinguish the concepts and think about the design of each concept as a feature. One concept is social zaps, public records of transactions. This implies a public balance of public transactions (Only an implicit feature for the on-chain zaps. currently basically not implemented for lightning ecash zaps). People are not currently concerned about the public balance of lightning zap transactions. Probably mostly because it's just not currently a feature. And maybe they will be more concerned about public balance in the future, whether its added as a feature or not. The other concept is private transactions. This is generally what people are trying to build tools for on-chain. Monero has more privatcy built in. (BTC was the first and simplest implementation of non-custodial VALUE. However that simplest implementation also made the transactions and balances public. But privacy is another important part of money and BTC didn't have a solution for that yet, initially depending on security through obscurity.) The public-private concepts also interact with the concept of privacy of transactions and balances for the sender and for the receiver. Some senders might want to send anonymously, and some receivers might want to receive anonymously.