NVMe devices using battery-backed DRAM or MRAM offer about 5µs access time, and current Optane SSDs are under 10µs. (NAND flash based SSDs with RAM-based write caches also have write latency below 10µs.)

So is the bottleneck really NVMe? Could there be a different protocol over PCIe that had lower latency?

NVMe was more or less designed to offer the lowest possible latency for a block storage protocol. You could beat it with direct memory mapping, but then you're limiting compatibility to systems with working 64-bit I/O addressing and storage media that doesn't require complicated management like NAND flash does.

You maybe consider about NVDIMM and persistent memory. But isn't available with consumer client products.