Except for Adi Shankara and Bhaskara, it is seen that all the Vedanta expounders are Vaishnavas...
First of all, Bhaskara's works clearly suggest he was a Vaishnava since he never refers Brahman to be anyone other than Sriman Narayana in his commentaries.
On the other hand, despite the modern Smartha Sampradaya's claims, many scholars have claimed Shankara too was a Vaishnava as most of his works which are undisputedly authentic, refer only to Sriman Narayana as Brahman.
Why aren't there exclusive Shaiva Vedantins?
It's a wide misconception that there are no Shaiva Vedantins at all. Yes, there have been Shaivite commentators on the Brahma Sutras, even though, they aren't as much as the number of Vaishnava commentators or even very popular. The two Shaivite commentators till date are:
Srikantha Shivacharya of the Shaiva Siddhanta sect, who subscribes to the philosophy of Shivadvaita (a philosophy very similar to Ramanujacharya's Vishishtadvaita). You can read his commentary on the Brahma Sutras, Srikantha Bhashya, in English, here.
Sripati Pandita of the Lingayat sect, who subscribes to the philosophy of Shakti Vishishtadvaita. You can read his commentary on the Brahma Sutras, Srikara Bhashya, in Sanskrit, here.
As for Agama, I am aware of Shaiva and Vaishnava Tantras, so this question doesn't relate to Agama.
Actually, the answer to your very question lies here. The primary reason why most Shaivite sects don't subscribe to the Vedanta school is because of Badarayana's criticism of Shaivite Agamas in Adhyaya 2 Pada 2 of the Brahma Sutras:
Topic-7: God Is Not a Mere Superintendent
For the Lord there can be no creatorship, for that leads to incongruity.
And (the incongruity arises) because of the impossibility of a relationship.
And (the position is untenable) because of the impossibility of (Nature) coming under (His) direction. (Or) And (God cannot be proved), since no physical support (adhisthana) is possible for Him.
Should it be argued that God will direct Nature like (a man directing) the organs, then it cannot be so, for that will result in God’s having experiences (of happiness, sorrow etc.). (Or) If a body, equipped with sense-organs, be assumed for God, (we say that) this is not possible; because of (consequent) experiences etc.
God will be subject to finitude or loss of omniscience.
As you can see above, almost all Shaivite Agamas see Brahman (Shiva) as only the efficient cause of the universe. In contrast, Badarayana argues that Brahman is both the efficient and material cause of the universe.
The only set of Shaivite Agamas which does accept Brahman to be both the efficient and material cause of the universe are the Bhairava Agamas, the principal texts of Kashmiri Shaivism. However, even Kashmiri Shaivites are staunch critiques of the Vedanta school because of Brahma Sutras' restrictions on lower castes and hybrid classes to Self-realisation as discussed by Badarayana in Brahma Sutras 1.3.38:
And on account of the prohibition, in Smriti, of (the Sûdras') hearing and studying (the Veda) and (knowing and performing) (Vedic) matters.
Hence, most Shaivites don't accept the Vedanta school for its caste-based restrictions and criticism of the Shaivite Agamas.
