The nonce you are hashing is done against many things, one of them is the coinbase transaction which must be in the block header, that transaction ensures the payout goes to the pool address, if you change anything in that transaction the hash you sumbit will be invalid to both the pool and the network as a whole.
The pool keeps all necessary data about block header and transactions for a valid submit to bitcoin network and miners are not given all data by the pool. So only the pool can submit solved block header which includes nonce. Transaction data, sometime, is a huge data; so it won't be given to miners in stratum mode. This answers your curious or doubt.
Mining on self node in BTC network has a fundamental good point is that a BTC is going to your BTC address, which can avoid a risk of operation failure in pool side or dishonourable operator of pools. So solo miner must bear this in mind.
Does this mean that the operator of a solo pool has the technical possibility of manipulation, e.g. to redirect the reward to himself in case of a block find? And this although the user of his pool had entered his own Bitcoin address as user name when started the mining? I would be interested from a technical point of view whether this would be possible in general. That would be disastrous, of course, because a pool operator could be a scammer. Can you please explain this in more detail? Thanks.