Suppose we have an response variable $Y$ that is an ordinal random variable with $1 < 2 < \dots < N$ classes. From wikipedia, one way to specify an ordered logit model is via the following generative model for $Y$, by assuming $Y$ is categorically distributed such that the probability that $Y$, conditional on covariates $\mathbf{x}$, falls in the $i$-th class or less is:
$$\Pr(Y \le i | \mathbf{x}) = \operatorname{expit}(\theta_i - \mathbf{w} \cdot \mathbf{x})$$
where $\operatorname{expit}$ is the cumulative distribution function of the logistic distribution, cut-points (thresholds) $\theta_i$, for $i = 1, 2, \dots, K-1$, covariates $\mathbf{x}$ and regression coefficients $\mathbf{w}$.
It seems to me that it is not necessary to include the constant in $\mathbf{w} \cdot \mathbf{x}$, that is:
$$\mathbf{w} \cdot \mathbf{x} = w_1 x_1 + w_2 x_2 + \dots$$
instead of
$$\mathbf{w} \cdot \mathbf{x} = w_0 + w_1 x_1 + w_2 x_2 + \dots$$
Indeed, the linked wikipedia section states that $\mathbf{w}$ and $\mathbf{x}$ are both vectors with equal length. The constant $w_0$ is not required as including one uniformly shifts the cut-points $\boldsymbol{\theta}$.
However, in the R package ordinal, the function clm, to fit the cumulative logit model states:
The model must have an intercept: attempts to remove one will lead to a warning and will be ignored.
Why is this necessary? Is it generally useful to include an intercept while fitting ordered logit models?
