Unsupervised Clustering high dimentional data not having estimation for K
I point out several things first:


You have an unsupervised problem so the first to know is don't search for The Right Answer. There is no right answer and you take the best you get acording to some pre-defined criteria!
not knowing K is another problem: This is one of many questions in unsupervised learning which do not have any answer! All those methods like elbow are just heuristics.
Kmeans doesn't report one of the nodes as center for the cluster: K-Means can be modified to do so. The simplest trick is to find the closest neighbor of final representatives from the data point and set them as representatives of clusters.
Some methods that you mentioned like K-Means and Sillouette assume gaussian clusters. If your clusters are not properly compact and well-separated, then they mislead you.


After these, let's have a look at your problem. Try to get an insight about your data with visualization techniques. tSNE is pretty sensitive to its parameters and needs more investigation. Start with PCA, KPCA or LLE to get some idea about data. In case they didn't give you a proper embedding, then go for UMAP. UMAP is also sensitive on parametrs but less than tSNE and it's pretty faster. It will give you more compact clusters if it finds any which is a bless for your work. At the end if none of them worked maybe try tuning tSNE parameters and see if you can find a good embedding.

On top of UMAP or tSNE you can try your clustering alorithm and it will be a logical pipeline for your problem.

Good Luck!