I don't think it reduces security much.  It is extremely unlikely that someone would publish an invalid block, since they would waste their POW.

Presuming that new headers are correct is reasonable, as long as you check the full block within a few minutes of receiving the header.

 

If anything, it increases security, since less hashing power is wasted while the full block is broadcast.

Block propagation could take the form

- broadcast new header

- all miners switch to mining empty blocks

- broadcast new block

- miners update to a block with transactions

If the block doesn't arrive within a timeout, then the miner could switch back to the old block.

This would mean that a few percent of empty blocks end up in the blockchain, but that doesn't do any harm.

It is only harmful, if it is used as a DOS attack on the network.

The empty blocks will only occur when 2 blocks are found in quick succession, so it doesn't have much affect on average time until 1 confirm.  Empty blocks are just as good for providing 1 of the 6 confirms needed too.