AFAICT (not authoritative, please correct me if wrong on the way it works here, my understanding is a touch shaky)
The 'mining' operation is basically a race to produce a valid block for the blockchain and have other miners accept it. The hash input data for the block is transaction data. Per-protocol rules, any miner is allowed to ignore any transaction they want to. Partly this is so that they can (if they wish) concentrate on processing transactions with a decent fee attached. This is thought to become a more significant thing as the built in reward for block discovery shrinks over time.
Therefore there is no guarantee that any transaction will be processed at all, and certainly no guarantee that it will be done in a timely fashion. Though it probably will. This is part of the wait for confirmation.
If a large miner group ignores satoshidice it doesn't mean that they disregard satoshidice transactions, it just means that they won't use them as input to the blocks they're attempting to mine. Others will still use them as input, and if the successful block comes from one of these others then the whole ecosystem counts the transaction.
However, If everyone did this, then satoshidice would indeed find it close to impossible to get transactions into the system.
--edit-- obviously this doesn't address your anonymity concerns, but others have done that.
The 'mining' operation is basically a race to produce a valid block for the blockchain and have other miners accept it. The hash input data for the block is transaction data. Per-protocol rules, any miner is allowed to ignore any transaction they want to. Partly this is so that they can (if they wish) concentrate on processing transactions with a decent fee attached. This is thought to become a more significant thing as the built in reward for block discovery shrinks over time.
Therefore there is no guarantee that any transaction will be processed at all, and certainly no guarantee that it will be done in a timely fashion. Though it probably will. This is part of the wait for confirmation.
If a large miner group ignores satoshidice it doesn't mean that they disregard satoshidice transactions, it just means that they won't use them as input to the blocks they're attempting to mine. Others will still use them as input, and if the successful block comes from one of these others then the whole ecosystem counts the transaction.
However, If everyone did this, then satoshidice would indeed find it close to impossible to get transactions into the system.
--edit-- obviously this doesn't address your anonymity concerns, but others have done that.