My understanding is that this is best done with a combined model that evaluates together both the probability of class membership along with the associations of class membership and covariate values with outcome. Larssen (Biometrics 60: 85–92, 2004) developed such an approach with categorical predictors of class membership. You maximize the overall (partial) likelihood of the data combining both the class-membership model and the survival model. That avoids problems that can arise from a 2-step model and its fixed assignments of potentially highly uncertain class memberships. As a more recent example, Liu et al. (Computational Statistics and Data Analysis 91: 40–50, 2015) extended this to arbitrary predictors and individual longitudinal data in parametric Weibull models, and note why a combined approach is better than a 2-step model.
I don't know of any off-the-shelf programs to do this combined class/survival analysis, however. Also, I understand that these approaches require a pre-specification of the number of classes. There might be Bayesian approaches that would also model the number of classes, but that's far beyond my expertise.
Besides the general cautions expressed in the post on latent class models linked in a comment by @rep_ho, make sure that the way you try "to predict the time-to-death of any new patients only based on these biological features" makes biological and clinical sense. These sound like cancer outcome data together with tumor gene-expression data, but even if they are something else the following principles still hold.
Make sure that the "biological features" include all the associated critical clinical characteristics.
First, there presumably is clinical assessment of the "different diseases" at question. That clinical assessment should be a primary factor in class assignment. Tumors originating in the same organ can have well defined subtypes, as in breast and lung cancer. You might want to start by assessing how gene expression values are associated with these major clinical distinctions among tumor types.
Second, each type of cancer also has its own further breakdown into stages specifically related to outcome for that type of cancer. Ignoring that information (either the overall staging, or the factors that go into defining the stages) is done at your peril.
Third, the same type of cancer can be treated with different therapies. For example: primary therapy of surgery versus chemo (and/or radio) therapy, radiation and/or chemotherapy adjuvant to surgery, classic systemic chemotherapy versus therapy "targeted" against particular molecular aspects of a tumor versus more recent immunotherapies. There is no reason to expect that molecular characteristics of a tumor will have the same associations with outcome for different therapies received even in a single tumor type. You might need to examine interactions of your thousands of biological characteristics with therapy in terms of outcome.
If you are using an approach like elastic net, you might best be served by not penalizing the critical clinical variables, reserving your variable selection and penalization for the other markers like gene-expression values.