Yes exactly, transaction verification has to be profitable, and the reward prevents double spends. Mining in Bitcoin is not profitable on average, but there is a competition. However the unsolved problem: how to create a system where supply (blocksize more or less) adapts to demand? It makes sense to have a fee market, but nobody really proposed something which explains how this might work, in my opinion. A system doesn't need 5000 nodes. 50-100 nodes have enough reliability. In Bitcoin what matters anyway is the hashpower, not number of nodes. Most of these things can be much better understood from the perspective of 2008, not 2010+. Satoshi wanted the system to be as open as possible to maximise chance of success. In 2016 it is much more clear where the problems are.
What do you mean by 'transaction verification'?
The word mining in Bitcoin mixes to issues: validation of transactions and creation of money. Also the word miners isn't compatible with a true Peer-to-Peer system. In any case, the important part is that doing the verification is profitable. Another word for the work these nodes do is auditing. It's the computer analog of a person checking the accounting statements in double entry book-keeping system. The blockchain is similar to the invention of double entry book-keeping by Luca Pacioli in 15th century Italy, and the use of clay tablets in Ancient Babylon to preserve contracts. Imagine a world without paper, how do you know that two people agreed on something ex-post? The invention of the blockchain allows for the first time in history to store completely immutable data. However when Bitcoin developers claim signatures can be stored independently of the blockchain, it might be that the Bitcoin blockchain is not that immutable after all. Overall the goal is to preserve some record which is completely indestructible, such that the record of ownership is completely secure. With fiat money not only are the ledgers in the hand of the nation state, but those ledgers are very hard to audit. Most people use fiat money every day, but the number of people who check the books of their central bank is small indeed. If one understands these things, it is clear that it is a certainty that in the future these public ledgers will be widely used.