Eventually yes, that would happen. Not right away because there is a lot of headroom of spam, crap, and low-value uses that will be forced off, ensuring that capacity is available, allowing ample time for improvements in capacity and throughput.
Fees rise, transactions stall, businesses, suddenly providing poor user experience go into reverse adoption. Nothing happens for months and months since there is no room for growth and then the halving happens, miners switch off and our transaction volume drops to 1/3 or less because you can't compensate for slower blocks by increasing the transactions per block and difficulty moves out to 30 days and then longer as declining value causes more miners to switch off.