Why does the wallet keep a pointer to the best chain When I run Bitcoin core to synchronize the blockchain without getting innvolved in any transaction, the wallet.dat gets written to nevertheless.

I believe this is the case because the wallet keeps track of the best chain. (However, I am not sure if this is the only change that is written to the wallet.)

Now, I am wondering why the wallet needs to keep track of the best chain. My possibly faulty understaning is that there is a consensus about the best chain which can be deduced from the blockchain. Hence, no seperate storing of it should be necessary.
I think it is necessary. The wallet doesn't just contain addresses. It also contains transactions that are sent to you. 

Suppose you generate a wallet, then back it up. You receive a bunch of money. You delete the wallet on your computer, and restore the old wallet. How does Bitcoin know that it should rescan the blockchain for transactions sent to you? I don't think it can, unless the wallet contains a field that represents the current best chaintip.

If you don't want Bitcoin to ever write to the wallet, you can compile it with --disable-wallet.