According to Confucianism, rituals serve mainly practical functions, as well as in some sense spiritual by promoting community spirit

• Not everyone finds benefit in practicing rituals, particularly if they associate them with a sense of religiosity which they do not possess or relate to.
• However, we can learn from the Confucian sense of ritual, Li 禮, that rituals are not necessarily religious, but rules of propriety, as well as positive habits and routines that serve to better ourselves and promote social harmony with others.
• Li is described as “a social medium through which we interact with others in order to achieve common values and to turn our society into a harmonious and aesthetic space”. -Nam Kyung-Hee
• Here, we see rituals have practical functions as well as immaterial ones.
• Engaging in rituals with other people solidifies community and communal identity, serving as social cement. This is in some sense a spiritual function of ritual as strengthening community spirit.
• Whilst not everyone may choose to engage in religious rituals, we all engage in social rituals as we are part of society and perform routine communal acts which serve to keep society functioning.

Read more:
Confucius - The Great Learning
Eno - The Analects of Confucius
Liu - The Self and Li in Confucianism

Along with Confucianism, it's interesting to look at rituals in the Buddhist practice, where they serve as tools for the progression towards enlightenment. Mental concentration and mindfulness are two of the most practical rituals that can carry over to anyone.

• Buddhism uses meditation and mental concentration as a foremost set of rituals for attaining enlightenment. Meditation and intensive mental concentration are means to transform the mind, to see existence in a different way. But ritualistic meditation is something a lot of people, who may or may not identify as Buddhist, take part in.

• Ritualistic meditation in a Buddhist context may be more extensive than "everyday meditation", but many people who meditate without necessarily identifying as Buddhist consider it to be a helpful practice and commit to it ritualistically. In this sense, the ritual of any sort of meditation and/or mental concentration could be said to still be fulfilling a deep spiritual function for humans, since the states that meditation/mental concentration promote can often be deeply altering of awareness---similar to, or perhaps even the same as, those shifts of perception that occur as or as part of some spiritual practice.

Read more about general rituals in Buddhism: https://www.insightmeditationcenter.org/books-articles/rituals-in-buddhism/

Read more about the connection between spiritualism and meditation:
https://mindworks.org/blog/what-is-spiritual-meditation/

For some, rituals can provide a deep spiritual function by bringing people closer to their religion and reaffirming their beliefs.

• Rituals are inherent in the spiritual belief system of many cultures.
• The purpose of rituals is to elevate one’s physical and emotional consciousness in order to achieve harmony, and in some cases spiritual ecstasy.
• Rituals are important to spirituality because the repetition of enacting each ritual can provide focal points for people, this could reduce distraction and provide a level of comfort.
• Rituals communicate or are intended to communicate to self, others or deities. They provide a spiritual function in conveying information regarding the commitments, beliefs and values of the individuals performing the ritual and link.

A religious ritual is any repetitive and patterned behaviour that is tied to a religious institution or belief.

Rituals may be performed individually or collectively during predetermined times, such as praying at specific times of the day, events like fasting or mourning rituals after death. Alternatively, rituals can also be performed sporadically; for example, praying at various times throughout the day.

According to the sociologist Mervin Verbit, religious rituals can provide a methodological measure of religiosity. To assess how far rituals can provide a spiritual function, Verbit claims they can be broken down into four dimensions; content, frequency, intensity and centrality and these factors may vary from ritual to ritual.

Change "spiritual" to "psychological" and I would agree. Everyone has their own interpretation of the word "spiritual", and if you were to refer it to a "spirit", well there is no evidence of such a thing. But yeah rituals have been demonstrated to have numerous psychological benefits, even in those that do not believe in the power of them.