Recently I've been seeing an increasing number of packages with labels such as "Contains no added nitrates or nitrites (except those naturally occurring in the celery juice powder)". Does celery powder have any use other than as a method of sneaking nitrates into things?
Yes, it tastes of celery. Celery is one of the three ingredients of mirepoix, the vegetable mix which is ubiquitous in French cuisine and has spread to many others. Current large-scale food production rarely includes slicing fresh vegetables into small cubes and browning them in a pan, but they try to add the ingredients in more convenient form. So, if you eat powdered soup or similar, it can be used for that.
On the other hand, if you are eating cured meat, then the most probable reason are indeed the nitrates. If you are looking at a certain type of cured meat (e.g. bacon), it is impossible to produce the same cured meat without nitrates *, but there are customers who are scared of "chemicals" and are more likely to buy the food if the label states that the nitrates are coming from a plant source.
* And in general, most types of curing are done with nitrates. There are types done without nitrates, e.g. the prosciutto pointed out in the comments, but they are the minority. And it is more difficult to create a safe process which cures without nitrates, since they have a preservative role.
As stated, celery powder is certainly used for flavor, in some applications.
But, as you are questioning, in cured items is has become in vogue to use it in an attempt to pretend that nitrates are not being used, which is misleading to say the least. In those items it is being used as a source of nitrates and frankly for marketing. I have seen multiple reports that bacon and ham for instance, sold as nitrate free but made with celery powder often are actually higher in nitrites which turn into nitrate than many made with refined nitrate and nitrite.
How can this be? Well, in the US, the USDA considered celery powder to be a flavor additive, not a preservative, even when the actual use is as a nitrate source. Frankly, any time the wording is similar to as you quoted, that no nitrates were added except naturally occurring..., then assume that is exactly the case, they added the item just for its nitrate content, not for its flavor or only marginally so.
It's used as one of the spices in KFC's Seasoning. (Celery Salt being a 3:1 ratio of salt and celery powder)
Idk if you've ever made fried food using their seasoned flour recipe, but I recommend it. I always have a bowl of KFC flour and it definitely enhances any chicken parm, croquette, etc, that I ever made.