What we find as Muktinath today, some 20 KM day-long trek from Jomsom, is very likely not the Saligramam Thirumangai Azhvar sang about. The actual Dhivya Desam must be further up the mountain, a week-long trek, where river Gandaki begins. In describing this dhiva desam Thirumangai Azhvar uses the phrase "சாளிக்கிராமம் அடைநெஞ்சே" instead of வணங்குதுமே or பணிமினே that he used for other dhiva desams. The only other divyadesam for which he use the phrase was Badinath -- "வதரி அடைநெஞ்சே". This may mean he did not really visit these two dhivya desams in person.
I visited Muktinath some 15 years ago. At that time there was a male priest, helped by his daughter. Perhaps this daughter has now taken over. The temple is encircled by 108 Go-Mukha figurines with water from Gandaki river diverted to flow through the mouths of the figurines. I remember taking a walk with the ice cold water falling on my head. By the time I reached the half way point the water falling on my head felt like a ton of bricks.
There is also a small minaret looking structure and a granite plaque just outside the temple walls. The plaque was engraved with a record of Bhagavat Ramanuja visiting the site. This structure is a modern placement.
There is a Thenkalai SV type Matam at a distance, not sure whether there are any Tamil speaking SVs there.
I am quite dubious that this place represents the true Saligramam that Thirumangai Azhvar sang about. It is probably a relatively modern invention. It serves very well as an exciting religious tourism spot for the general public and a place to make a decent livelihood for the priest there, male or female.
Cheers!