Can we drive the bitcoin transaction fee arbitrarily close to zero by keep increasing the block size limit? Can the bitcoin scalability problem be solved in the long run by increasing the block size limit (either dynamically or statically)?
No. The fees are the incentive for the miners to continue their work, verifying transactions and including them in blocks. Mining is expensive. Eventually, the block reward will reduce to zero and the fees will be the only payment that the miners will receive.

Whatever may change with Bitcoin in the future, the fees must be able to float to the market and that must be protected, although, I do disagree that the current auction model for limited bandwidth is the optimum model for Bitcoin to be using as it is nowhere near scalable.