Do you not also want that $\mathbb{P}_Y$ is $\phi$-invariant?
In any case, yes there are extremely many continuous-time continuous-path real-valued stochastic processes whose law is ergodic under the time-$1$-shift map. Of course, the most trivial example would be where
$$ \mathbb{P}(Y_t=c \ \ \forall t \geq 0) = 1 $$
for some $c \in \mathbb{R}$. An almost-as-trivial case would be
$$ Y_t = p(2\pi ft + \Phi) $$
where $p(\cdot)$ is a $1$-periodic continuous function, $f$ is an irrational number, and $\Phi$ is a random variable with $\Phi \sim \mathrm{Uniform}(0,2\pi)$. (I don't know if this would count as a "non-trivial example class" under your bounty statement!)
Perhaps the simplest "non-trivial" example would be the Ornstein-Uhlenbeck process:
(I hope there are no typos in the following!) Fix two parameters $\theta,\sigma>0$. Starting off with a two-sided Wiener process $(W_t)_{t \in \mathbb{R}}$ (meaning that $(W_t)_{t \geq 0}$ and $(W_t)_{t \leq 0}$ are independent Wiener processes), we can define a stochastic process $(Y_t)_{t \in \mathbb{R}}$ where for almost every sample path of the Wiener process, for all $t \in \mathbb{R}$,
\begin{align*}
Y_t &= \sigma \! \int_{-\infty}^t e^{\theta(s-t)} \, dW_s \quad\quad\quad\quad\quad\quad\ \, \text{(Riemann-Stieltjes integral)} \\
&= \sigma \left( W_t - \theta \! \int_{-\infty}^t e^{\theta(s-t)} W_s \, ds \right) \quad \text{(classical integral, obtained by integration by parts)}.
\end{align*}
This is the unique two-sided-time strong solution of the stochastic differential equation
$$ dY_t = -\theta Y_t \, dt + \sigma dW_t, $$
or equivalently (as a random differential equation)
$$ \frac{d(Y_t - \sigma W_t)}{dt} = -\theta Y_t. $$
$\text{[}$The one-sided-time solution $(Y_t^{(y_0)})_{t \geq 0}$ of this equation starting at any $y_0 \in \mathbb{R}$ would be
\begin{align*}
Y_t^{(y_0)} &= e^{-\theta t} \left( y_0 + \sigma \! \int_0^t e^{\theta s} \, dW_s \right) \\
&= \sigma W_t + e^{-\theta t} \left( y_0 - \sigma\theta \! \int_0^t e^{\theta s} W_s \, ds \right). \text{]}
\end{align*}
A one-sided-time stochastic process with the law of $(Y_t)_{t \geq 0}$ (or a two-sided-time stochastic process with the law of $(Y_t)_{t \in \mathbb{R}}$) is called an Ornstein-Uhlenbeck process.
Sketch-proof that OU process is shift-ergodic. Firstly, working in two-sided-time,
- one can check that replacing $(W_t)_{t \in \mathbb{R}}$ with $(W_{t+1}-W_1)_{t \in \mathbb{R}}$ transforms $(Y_t)_{t \in \mathbb{R}}$ to $(Y_{t+1})_{t \in \mathbb{R}}$;
- since the Wiener process has memoryless stationary increments, one can show that the law of $(W_t)_{t \in \mathbb{R}}$ is mixing and hence ergodic with respect to the map $(\gamma_t) \mapsto (\gamma_{t+1}-\gamma_1)$;
and so the law of $(Y_t)_{t \in \mathbb{R}}$ is ergodic with respect to the map $(\gamma_t) \mapsto (\gamma_{t+1})$. Finally, having ergodicity in two-sided time guarantees ergodicity in the one-sided-time restriction.
In general, given any $\phi$-invariant probability measure $\mathbb{P}$ on $\mathscr{C}$, one can express $\mathbb{P}$ as a weighted average of ergodic $\phi$-invariant probability measures on $\mathscr{C}$. This is essentially the ergodic decomposition theorem, which holds for any Borel-measurable dynamical system on a Polish space; in this case, the $\sigma$-algebra $\mathfrak{C}$ is precisely the Borel $\sigma$-algebra of the topology of uniform convergence on compact sets, which is a Polish topology.
