Even if the block space is available there is noting that says each block must be "filled" to the max block size. Even if the max block size is large, miners to have a small incremental cost for including each additional TX in their found block so they would likely not wish to include a lot of no fee TXs
Miners may have a small marginal cost of including a transaction and therefore there is likely to be some low fee level. However, in a competitive mining industry and unrestricted market, price will equal marginal cost. At this level the equilibrium difficulty level may be too low. We need artificial restrictions on the market to increase fees and therefore have a higher equilibrium difficulty.
The marginal cost increases as the miners' blocks get bigger as the chances of their block getting orphaned will increase as the miners' blocks get bigger. I would also say that regardless of the block size, the number of no fee TXs will decrease as the block subsidy decreases as the miners are generally not going to want to confirm zero fee TXs when they are receiving a diminishing amount of revenue