That really depends on the game. Loading every texture in advance isn't practical for all games - many newer "open world" games will stream textures to the GPU as needed based on the player's location and what they're doing.

Also, modern game textures are a lot of data.

True, on-demand loading/unloading of large textures still needs to be handled. Video streaming handles congestion by sacrificing fidelity to reduce bitrate. A similar approach could be taken with textures by downsampling them (or, better yet, streaming them with a compression codec that supports progressive decoding).