I'll take a stab at this. I worked in the masonry restoration biz for a long time in CA. I am not a mason but have had to address such issues as yours. I am giving you an opinion based on my experiences in the field. Cementitious or non-cementitious grout, that is the question...
Cementitious has been around forever. It is easier to clean up (water). It will be affected by freeze/thaw cycles (if that's pertinent to your location), the environment, the chemicals in the pool, shifting of the earth. Easily removed and replaced. If tinted a color, that color will change and fade with the sun and chemical exposure. NOTE...if tile abuts concrete or wood, or stone (any dissimilar material), that must be an expansion joint filled with the appropriate materials that will expand and contract.
Non-cementitious grout (epoxy, quartz for ex.) has been around a while and is always being improved. Harder to work within the application and clean-up (epoxy) (quartz less so and cleans up with water if memory serves). these materials are more flexible than grout, slightly so. They won't stain and will resist water and chemicals well. If a colored variety then that color will fade in time. Replacement or repair is similar to grout, and with proper prep, can be used to repair incremental damages. Note. test material first. epoxies can darken adjacent stone, tile, and concrete.
Yes, it takes more time to use/cleanup the epoxies or quartz materials. the cost may be justified if looking long term towards maintenance. Try negotiating the price (T&M).