Wouldn't you get an arms race resulting in ever increasing security, while at some point extra security would be redundant? If miners are now paid for adding security, wouldn't it make more sense to pay them for adding speed once security is at a high enough level?
I think you have a misunderstanding of how Bitcoin works.
You can't add transaction speed. When you make a transaction on the core Bitcoin network it gets immediately broadcast and (if valid) relayed to the rest of the nodes participating in Bitcoin. This happens nearly instantly. So if you send me one of your bitcoins I'll get that in my wallet quickly whether I'm sitting right next to you or across the world. The problem is knowing if I can trust that the transaction will be seen as valid by the network forever into the future. To know that I need to wait for what's called "confirmations".
Confirmations happen when the transaction is included into a block by a miner on the network. It is widely held that after 6 confirmations (6 blocks that acknowledge the transaction) it can be regarded as permanently valid because to undo that requires immense computing power, and more so with further confirmations.
The time between Bitcoin blocks is automatically adjusted to remain at a target of 10 minutes to keep the number of coins coming into circulation relatively steady. So to have a fully confirmed Bitcoin transaction takes on average 1 hour on the core network. This is another reason I believe off-chain transactions, which can be confirmed as valid instantly, will play a big role in Bitcoin's future.
The security of the network (difficulty in attacking it) automatically rises or lowers depending how many miners participate. As Bitcoin adoption grows the number of mining participants, and therefore security, naturally rises.