What you recognised is a common issue and it occasionally manifests into situations where people are perplex as to "why am I predicting flat values?".
CV.SE has some very enlightening topics on this matter in: Why I get the same predict value in Arima model? and Flat Forecast from ARIMA and SARIMA.
Let's take as an example a simple time-series model, like a first order auto-regressive model AR(1), where $y_t = \beta_0 + \beta_1 y_{t-1} + \epsilon_t$ and $\epsilon_t \sim N(0, \sigma_\epsilon^2)$. In this case our estimates $\hat{y_t}$ are simply $\hat{y_t} = \hat{\beta_0} + \hat{\beta_1} y_{t-1}$ because $\epsilon_t$ is expected to be zero. Nevertheless as we extrapolate $y_{t-1}$ has to be itself estimated because it is unavailable. This leads to situation where after some point, we actually use our own predictions are input data.
The fact that "we use our own predictions as inputs" is epitomised by seeing that certain time-series algorithms are presented under a filtering approach, the Kalman filter and the Holt-Winters filter being prime and widely used examples.
So to become particular to what was originally mentioned: if we want to create our own forecasting routine that does not simply offer one-step-ahead forecast we need to be able to be populate our "lagged features" with their predicted values. That's why most forecasting routines (e.g. forecast::forecast, smooth::forecast, prophet::make_future_dataframe, bsts::predict, KFKSDS::predict, etc.) have an explicit horizon, periods, n.ahead, etc. argument. We need to know how far we look into the future to appropriate update/populate our beliefs to get there!
