Bitcoin should absolutely not be sending the computer’s time to the network. Nor should it depend on the clock being set properly. Each node can keep track of when they see transactions come through. If they want to reject a block because it contains a transaction they saw too long ago, then they are free to reject it; no reason for the nodes to compare each other’s clocks. If it’s too old they will all reject it, which is what we want.
If someone rejects a block that most of the network accepts, then all of the blocks that they produce will be invalid and transactions won’t gain confirmations from their perspective.
This is about times of blocks, not transactions. Block timestamps are used for the difficulty calculation, so they can’t be too far off from reality.