Smudging is an English word and its reach is the first problem. It is used for Lakota practice with sage and sweetgrass, for Dine practice, for Anishinaabe ceremony, for Cree and Blackfoot, and by extension for anything anyone anywhere burns for cleansing. These are not one practice. They differ in plant, in occasion, in who may conduct them, in what is being asked, and in whether they may be described to an outsider at all. A single English noun holding them together is doing damage before anything else has been said.
The commercial article is easy to describe because it is standardised. A bundle of white sage, often packaged with a feather and an abalone shell, sold at a fixed price with a card of instructions, generally carrying the words cleanse, negativity and energy, and generally naming no nation at all. That object is a twentieth-century assembly. The shell and feather combination in particular has no single traditional source and is best understood as a retail convention.
The plant is a separate problem. Salvia apiana is a shrub of limited range in southern California and northern Baja California. It is not the culinary sage of a European kitchen garden and it is not interchangeable with it. Commercial harvesting for the wellness market has drawn documented objection from Indigenous Californian communities on three distinct grounds: access to traditional gathering sites, unpermitted harvesting on public and private land, and the condition of the stands themselves. Suppliers who sell cultivated rather than wild-gathered stock now advertise the fact, which is a fair sign that the concern is real rather than rhetorical.
On the central question this entry takes the position the site takes elsewhere. Several of the ceremonies grouped under this word are held by their nations to be closed, and those nations have said so in public statements under their own names. That a full account of one of them can be found in an ethnography, a memoir or a website is not a permission. Material published without consent a century ago does not become available because it is old.
It should also be said that Native opinion is not uniform. Some practitioners use the English word for their own ceremonies without difficulty and teach within their own communities; others regard the word itself as an imposition that flattens things which ought to be kept apart. Both are positions held by people with standing to hold them, and an outsider reporting the disagreement is not obliged to settle it.