This is not generally true. A sudden increase in mining hash power (more miners) might temporarily distort the confirmation time so that is is faster for a short while, but within 2016 blocks the difficulty will be adjusted and the average confirmation time will return to 10 minutes per block.
So, more miners means more difficult mining, but not faster transaction verifications.
Also, there is a detail that is confusing the way you describe it.
You state:
"Miners get paid twice, once for verifying the transactions and again when they successfully generate new bitcoins."
This makes it sound like a miner gets paid for verifying the transactions even when they don't successfully generate new bitcoins, when the two things are actually tightly tied together. While the miner gets paid from two different sources (a block subsidy and transaction fees), that is handled with a single payment when "they successfully generate new bitcoins". So it would be more accurate (and less confusing) to state that upon successfully solving the mining calculation, miners receive a single payment from two sources (from the transaction fees, and from the block subsidy).
Thanks for clarifying that. I'm a bit of a newbie myself to Bitcoin mining so it's good to know.
