Vinyl or laminate tiles that look like Saltillo ceramic tiles

Yep! Back then they used luan plywood over subfloors to level them and a smooth base for tile. Sheet rock does not go on floors. What they use now is cement board over the subfloor. You would not have a problem with grout drying out. Thinset is used to adhere ceramic or stone tile. Grout just seals the joints. It never comes in contact with the subfloor. As long as the plywood is flat and stable you still can tile over it or you can install cement board over it and tile. One thing to consider is cement board and the tile will raise the height of the floor.

https://www.armstrongflooring.com/residential/en-us/flooring-installation-advice/articles-videos/vinyl-flooring-that-looks-like-stone.html

You should not have a problem as long as the subfloor is level and stable. I do suggest checking for any movement and screwing down any loose or flexing area. Main problem with your neighbor if the floor was not solid and stable. Movement in any subfloor whether plywood or cement board will cause the grout to crack and fall out. Another problem is if the grout was not a premixed type it could have been mixed improperly. Wrong amount of water, not allowed to "slake". "Slaking" is letting the grout sit for a few minutes after mixing so each grain absorbs moisture. It develops a slight skin on top. Similar to letting a steak, roast, chops, chicken, sit for about 5 to 10 minutes after cooking to let all the juices settle and spread out for a tender piece of meat. My daughter had ceramic tile installed over vinyl tiles in her bathroom. I was skeptical when she told me. I wouldn't do it. It's been over four years and still holding.