Professionals messing with water systems are always well overcosted for the seeming scope of work. The reason being that they need to take out the pan, the walls, some tiles, inspect the floor around the area, remove and replace whatever seems like it’s been damaged, find replacement tiles that are either the same or similar… and the most important part is that they make sure that if they have to come back in 5-30 years for any leaks, they want that to be covered up front as well. You’re effectively paying for leak insurance from them for the next however many years they offer, as well as a potentially brand new bathroom built if the damage is bad enough.

You want a one day fix for a singular part of a greater (and notably damaged) area. They want to make sure they do it once, and then don’t ever hear from you again for that particular part. So there’s all of that and quotes apply Murphy’s Law for unseen issues. The whole floor may need to be replaced, every structural member under the floor might be weakened from excessive water damage and needs either replacement or support, if you have tiles on the walls, some of the walls may need replacement, if there’s an exterior wall involved there they might need to check the those structural members for damage and then possibly replace them, etc. It extends to the whole room and a bit beyond.

I’m going to guess the 16k quote was a “I don’t want to do this, so price multiplied to assure I don’t need to”. My only suggestion is to ask around among others you know if they’ve had work done and by who. There are plenty of freelancers out there who will do exactly the one part you want to do and charge (based on current info) somewhere around 1.5-3k to do just the part of the job you want done, with none of the protections from the larger companies aside from immediate leak issues. Finding a competent freelancer is the toughest part, which is why I always suggest getting one from someone else who has had a good experience (or more).