Will BCH inevitably run into the same problems as it grows to BTC size?
No.
Your limits are not reality's limits.
Once all valid criticisms or pieces of propaganda against BCH have been debunked or addressed, BTC advocates sometimes fall back to a last resort dismissal of "Well BCH is just too small to have problems, everything is easy when you're small. If BCH got to be the size of BTC it would end up with the same (or worse) problems."
This argument can be summarised as "We couldn't make our version of Bitcoin work as it grew, therefore it's impossible you can make yours work." Clearly, this is logically ridiculous & rather arrogant to boot. By definition, BCH can only objectively disprove this once it is the same size or larger size, but there are several critical distinctions that make it highly unlikely BCH will be stalled by any of the roadblocks BTC has:
- Radical differences: The entire reason for the BTC/BCH split was how irreconcilable the competing visions became during The Blocksize War. This initial division has lead to BCH & BTC being radically & increasingly different today. Dismissing the BCH community as doomed to repeat the mistakes of the BTC community ignores the large & intentional gulf between the projects & their history/methodology.
- Concurrent history: The BCH community is not operating in a parallel universe. The BCH community lived through the BTC history as well as their own, so they've had the chance to keenly observe & adapt to issues seen on the BTC side ahead of time.
- Resistance to Subversion: Unlike the BTC side, the BCH community would not tolerate its project being hijacked with propaganda & censorship. This means that not only were a huge array of false beliefs never implanted into the community to begin with, but BCH supporters are aware of & resistant to further disinformation attacks that are still easily employed on unwitting BTC advocates.
- Technical superiority: As a result of propaganda, BTC advocates believe their predicament of high-fees & an unreliable network is an inevitable result of their (relatively modest) network throughput. The reality is that even with BTC levels of traffic, BCH fees will not rise & network performance will not degrade. Furthermore, the BCH community has pre-emptively fortified itself against a repeat of any attempts at obstructing successful scaling with ABLA.
- Improved governance: One common possible roadblock mentioned is the difficulties BTC has struggled with in governance & upgrades. Although community size plays into this, so does propaganda, culture, history & more. Considering the example of increasing the blocksize, it is very obvious that BCH & BTC have entirely different governance structures & culture so it is far from a given that the struggles of BTC would inevitably affect BCH.
- Overwhelming Passion: The BCH community is not playing around. It will not rebrand & it fought two further civil wars even AFTER separating from BTC to keep the project alive. The BCH community brings that same resilience & dedication to every element of its one chance to become global reserve currency. The BTC community are undoubtedly passionate about their project, but they are not remotely comparable to the BCH community in terms of their determination, creativity or experience in keeping the real Bitcoin mission alive. Whether it be price declines, never-ending slander or sabotage, the BCH community has provably persisted through difficulties & pressure that the BTC community would simply crumple underneath (as the Blocksize War demonstrated, ending their realistic chance of reaching global adoption at the first life-or-death hurdle).
Once all these factors are taken into account, it is hardly conceivable BCH will be stalled simply as a matter of scaling up to the size BTC has. The project may struggle or stall at some previously unseen roadblock but it would almost certainly be a new challenge unique to the BCH project itself (not encountered by BTC or BCH previously, as those are all avoidable).
Note on Reverse Argument
Note, the opposite argument is also sometimes suggested - not that BCH will run into the problems of BTC, but that BTC can easily copy any advantages of BCH if it needs to. This is not going to happen either, see If BCH improved on BTC by raising the 1MB block size limit, what if BTC does the same?.
See also: Why is there so much propaganda in BTC?
See also: What's wrong with Bitcoin BTC?