Earlier quoted context omitted.
We are getting there. In-vitro methods, often involving cell cultures, are gaining traction. In-silico, also. Robotic automation and synthetic biology help with that a lot. Machine learning can help make sense of more limited data, for example predicting toxicity and other parameters for drugs so that you can better choose what to forward to Human trials.
In vitro studies have been done for ages, but are often a worse model than closely related organisms. For instance, there are various human cancer cell lines and while these are routinely used to study various aspects of biology (cancer being just one of them), they are generally a poor model for drug trials. The current research trend goes towards developing organoids — cell cultures with a distinct 3D organisation…
Nothing can replace certain studies in Humans completely. We can only hope to make those more effective by better choice of candidate drugs.