If you have your private keys, you are set. You can use your private keys to compute your Bitcoin address' and as you said all of the transaction information is stored provably in the blockchain.
You technically do not need your wallet file. You can import your private keys directly into your wallet software without the wallet.dat file. Your wallet.dat file is what holds your private keys you can always hold onto your private keys and import them when you need to spend your BTC.
Note however, that you better be absolutely certain that you really understand what a private key is and how the wallet works.
There have been people who thought that their passphrase was their "private key". Later when they went to recover their wallet, and they discovered that they misunderstood what a private key is, they were disappointed to learn that they would never again have access to the bitcoins.
There have also been people who had some of their private keys, but didn't realize that the wallet creates additional addresses without telling you about it. It keeps these addresses hidden. They didn't have the private keys of these additional addresses, and therefore permanently lost access to some of their bitcoins.
As long as you completely understand exactly what you are doing, and are very careful to do it correctly, it is sufficient to just have the private keys. In that case you do not need the wallet.dat.
If there is any chance at all that you might mis-understand something, or make a mistake, then it is better to have the wallet.dat (since the wallet.dat has all the private keys).
