Kalki avatar is an avatar of Lord Vishnu only. There is no doubt in this.
It has been prophesied in many puranas that Lord Vishnu will incarnate to remove evil from Kaliyuga therefore restoring dharma again.
In this Kali Yuga itself, when the junction period is yet to lapse, the lord will be born under the name of Kalki and Viṣṇu Yaśā. He will be born in the family of Parashara and have great exploits. [Brahmanda Purana Upodhgata Parva Chapter 73 verse 104].
Above verse is talking about Lord Vishnu and his avataras in different manvantaras.
From Bhagavata Purana:
Lord Viṣṇu — the Supreme Personality of Godhead, the spiritual master of all moving and nonmoving living beings, and the Supreme Soul of all — takes birth to protect the principles of religion and to relieve His saintly devotees from the reactions of material work. [SB 12.2.17]
Lord Kalki will appear in the home of the most eminent brāhmaṇa of Śambhala village, the great soul Viṣṇuyaśā.[SB 12.2.18]
There are many other references in puranas and in the Mahabharata it is said that Lord Vishnu incarnates in the sandhi period of Kaliyuga and establishes the conditions of Kruta Yuga. There are stotras found in the puranas which include Kalki in Dashavatara (tem important incarnations) list.
Ashwathama, Samba etc., are ansa-avatars of Siva and are responsible for the destruction.
Arjuna caused more destruction than both these. On 14th day of Kurukshetra alone, Arjuna killed an akshauhini while searching for Jayadhratha. Bhishma killed many soldiers in the Pandava side that Yudhishtira was scared and thought they would lose the war if Bhishma continued to fight till the last day. So, these were also responsible for destruction. But these are not avatars of Lord Shiva. If we look into Mahabharata in critical way, many characters like Karna, Drona can be made the responsibility for the great Kurukshetra war. These are not avatars of Lord Shiva. So, the logic saying Kalki being avatar of Shiva because he will kill evil people doesn;t work.
Lord Vishnu takes birth out of his own will to bring back dharma and remove adharma time to time. From Bhagavad Gita 4.7
yadā yadā hi dharmasya glānir bhavati bhārata
abhyutthānam adharmasya tadātmānaḿ sṛjāmy aham
paritrāṇāya sādhūnāḿ vināśāya ca duṣkṛtām
dharma-saḿsthāpanārthāya sambhavāmi yuge yuge
Whenever and wherever there is a decline in religious practice, O descendant of Bharata, and a predominant rise of irreligion—at that time I descend Myself.
To deliver the pious and to annihilate the miscreants, as well as to reestablish the principles of religion, I Myself appear, millennium after millennium.
[The translator used religious but the word is Dharma and it is untranslatable]
Hence, there is no doubt that Kalki is Lord Vishnu's avatar.