I actually consider the lack of Wi-Fi a feature.

I couldn't find a scenario where Wi-Fi would add value as opposed to BLE. It consumes more power, forces you to use TCP/IP, which pulls in lots of complexity without great solutions (you need to deal with pairing, security, etc), is more complex to set up... I could go on. Solutions like Thread are relatively recent (in fact I still don't know if licensing has become more reasonable).

BLE and Bluetooth Mesh give you tons of functionality with low power, in a complete solution. Do I really need my devices to have an IP address?

I guess this depends on what you're building, but in most cases this is about home automation, sensors, lighting, toys, etc.

Say I'm doing home automation and I decide to go with BLE and Bluetooth Mesh (I didn't realize this was a thing). What's the recommended chip/device/design to bridge this Bluetooth network with "regular" computers?

I can't speak to BLE/Mesh, but the Thread protocol (which Nordic also supports) has the concept of a "Border Router". Which is exactly what you're looking for.

Can anyone here comment on espressif's ble mesh implementation?

I saw a few demos d went through their sdk samples a year and a half ago but those were pretty raw then

I'm planning to use a BLE interface plugged into a Linux computer (RPi or something similar).