Bitcoin has the same problem once there is no more coin to mine. Transaction fees have to increase with both bitcoin and bitcoin cash in the long term.
False argument. Once fresh coin supply ends, Bitcoin Cash with its big block structure either loses hash rate or saturates blocks with cheap transactions, in the later case it eliminates small miners and falls in the centralization hole.
For bitcoin, it is a whole different story as it needs to develop its fee market for high value transactions for surviving without escalating the centralization problem.
Don't get me wrong, it is not good news for Bitcoin to lose the small transaction market segment, but it is not a catastrophic threat, and we have time for dealing with it, actually, this trend enforces more scaling solutions to emerge either minor improvements like Taproot and Schnorr signatures or more disruptive and radical ones (hopefully).
On the other side of the island, in BCH camp they do not
feel any pressure for smart scalability solutions because they are ok with large blocks and small fees, totally ignoring the centralization consequences of this scenario.