How to deal with a biased feature in Machine Learning (date)
What you are describing are censored observations in survival analysis. Indeed, those who are still alive will obviously die but just at a later, observable date in a time frame past our study date. By counting them as survived on a logistic regression we will clearly bias our model to favor survivals for those particular horses, unrealistically.

Hence, you have survival data and should approach it like so. The Cox proportional hazards rate model allows you to relate hazard rate functions to predictors, and while many use linear models for the "predictor" part there are other ML methods like xgboost and random forests that have the ability to return Cox PH models. These models allow for censored observations as you describe.