Why do recipes often say "Salt and pepper to taste" instead of "Salt and pepper at table"?

Because adding salt and pepper while cooking yields a different result than adding salt and pepper after serving.

There are a couple of reasons. Adding salt and pepper during the cooking process allows time for their flavors to infuse with the rest of the ingredients. If a dish goes unseasoned until it is served, then it will taste differently. Imagine having a piece of meat that was unseasoned before cooking. Sprinkle some salt on it, and it tastes like salty meat.

It can cause certain foods to cook more quickly by drawing out moisture and soften them, like when cooking carrots.

As a side note, I find that using sea salt in cooking vs. table salt (iodized or not) also makes a difference.

To some extent it’s chef’s choice what the dish should taste like, and the amount of salt is part of that. And @dappled_leaves makes a good point that it’s different if it’s added during cooking to how it is at the table.
I’d hope a chef would hold back on the salt and allow some adjustments at the table and anyone on a low-sodium diet can avoid that risk.

Personally I’m allergic to pepper and would hope that no more than a teeny bit is added in the kitchen.

I didn’t suggest eliminating the salt completely from the cooking process.

“Why would you salt and pepper to your taste if you’re cooking a meal that will be shared by others?
Usually I add minimal salt, and I add a bit of white pepper as often as I do the more obvious black.”

The question is about “to taste”. Who’s taste? If I’m writing a recipe for others, wouldn’t it be best to include the amount of salt I added to make the dish begin to taste as I want it to, then acknowledge that it may take adjusting at the table?

Edit:
I use sea salt, too, @Pied_Pfeffer, in a grinder, and you need much, much less if you add it to the finished product.

Putting some seasoning in the food while it is being prepared makes a difference to the flavor that seasoning at the table can’t compensate for.

^^^ @marinelife.

Saying “salt and pepper to taste” doesn’t necessarily mean the chef’s taste. All the instruction means is “salt and pepper are recommended; but unlike the previous steps, it’s not required and the amount you will need is far less exact.” When food is salted and peppered at the table, it just means there are a few more people in on the final recipe.

Nice question. Well, I’m no chef—in fact, I barely cook—but I think it could mean one, two or both of these things: One, since chefs can’t season their recipes to totally please everyone’s taste, they must simply please their own. And two, “to taste” may simply mean spicing up the recipe in general rather than to a specific preference.