First, of all, this issue has nothing to do with if you like/hate Lightning or if you liked/hated XIII. So please don't argue she should be in there because you liked her or she shouldn't because you don't like XIII. It simply doesn't matter.
Here's my thoughts on the whole matter... (bold is my points, non-bold is just supplementary)
Each new Final Fantasy accomplishes two things, 1-How it develops it's own identity and 2-How it further develops the overall Final Fantasy identity. EVERYTHING in each game must do one of these.
The overall identity of Final Fantasy is reiterated and developed through the use of recurring themes (light/dark, chaos/harmony, crystals), recurring character names (Cid, Biggs/Wedge) and other things like Chocobos/Moogles, Summons, weapons (Excalibur, Masamune), items (potions, antidotes, elixer) and battle elements (job system).
At the same time, certain things in each game establishes the unique world of each game. Primarily this is the story (with it's specific characters), story weapons (buster sword), game systems (FF8's draw system), etc.
Fundamentally by nature, a single thing CAN NOT do both - something that makes one game unique can't develop the overall series without detracting from the uniqueness in the original game. Likewise, something that appears in every FF cannot make a single game unique from others.
There are some things which may not seem as clearly black and white, but this still stands. For example, simply having Moogles in the game adds to the overall FF feel. However, each race of Moogle is different in each game. Therefore, XIV's Moogles add to the uniqueness of XIV's world. If it was the same race of Moogles in every FF, it would add to the overall FF series and NOT add uniqueness to XIV.
Let's take the draw system from FF8. This is something that is currently only in 8, therefore it is an identifying part of FF8. If it is used in another game, it is no longer a FF8 battle system. It is a Final Fantasy battle system that is used in 2 games, which includes 8. Similarly, materia used to be a FF7 game system. Now, it's in 2 games and the name can be reused in future games as long as the mechanics are similar. It's essentially turned into the job-system, a recurring FF thing slightly different in each game it appears in.
Whenever you take items that are unique to one game and add it to another for nostalgia, you are increasing a homogenous feel to the overall FF series, but at the cost of sacrificing the original uniqueness of said item and game. This is what we call fanservice. The degree of this depends on how the fanservice is done. So the ultimate question becomes "What is the right balance? How much uniqueness are we willing to sacrifice for fan-service?"
We finally arrive at Lightning in XIV, the fanservice in question. For me, directly placing an actual story character into another universe is one of the BIGGEST ways to detract the uniqueness of the original game for the sake of fanservice and promotion. This stays true no matter how well it is written into the lore and story. These effects can be "minimized" or lessened by saying it's just a temporary seasonal event where the actual lore of the game is "suspended." However even then, by simply being there it's detracting from both XIII and XIV's uniqueness to a degree.
Gilgamesh is often noted as being a character that does this, so therefore it should be ok for Lightning to cross-game travel as well. But people fail to think that having other characters also do this detracts from the uniqueness of Gilgamesh as well. The more you "pull out" the dimensional rift card the less unique it gets, like a joke told too many times.
In the end, everyone has different opinions on what the right balance is and it's just that - opinions. But everyone should be aware of this question of balance when they ask themselves what they're willing to sacrifice when Lightning is in the game, as well as moving forward in XIV with future fan-service.
For the record, here's my opinion: I don't like Lightning in XIV because I think it sacrifices too much uniqueness from both XIII and XIV. It's much better if it's a temporary seasonal type event like I've heard where the lore of XIV is "suspended." Regardless, while I'm reluctantly okay with the clothes and weapons, having her actually appear is a bit too much for me.