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Does this mean that the network becomes more cost-effective to operate as it grows?
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It means that the network is becoming cheaper to attack, basically. Satoshi was working from the assumption that the cost of mining will approach the price of the coins mined. If the TX fees don't increase substantially when block reward is halved, interesting stuff will happen: Hangers full of mining gear will suddenly become unprofitable to mine with -> available dirt-cheap for more creative use, like 51% attack.
No. Total fees paid to miners tend to
increase as the number of transactions per day increases.

Thus it would seem that the network simultaneously grows both
stronger and more cost-effective as the number of transactions per day increases.