tl;dr: Adjuvants may increase the response of the immune system if given without an antigen, but that immune response would not be directed towards the existing pathogen.

The reason behind this is tied to how the immune system targets a pathogen in the first place. The initial step is chopping up the pathogen/virus and creating antigens by the first line responders (known as antigen presenting cells, or APCs). These cells then give two signals to other white blood cells to induce the immune response: the antigen and the biochemical signals (cytokines) to teach the WBCs how to break down the pathogen. Without both signals, the antigen AND the cytokines, there is no further immune response.

Some adjuvants (aluminum/M59) increase antigen processing while other adjuvants artificially provide instructions on how to combat the pathogen. Without an antigen associated with the adjuvant, the immune system would fail to create any immune response to target the pathogen even if it's already in the body. Failure in this two-signal process is mirrored in some inherited auto-immune deficiencies as well as immune evasion of cancer cells.

If you're interested in a relatively recent update on how adjuvants work, this literature review go over recent advancements.

There are some great answers here, just adding a bit based off your recent edit.

"Evolution" is a great way to frame development of cancer cell lineages. For a cell to develop into malignant growth, it has to be able to "outcompete" neighboring cells to go from cellular expansion > avoidance of immune system > invasion of extracellular matrix > treatment resistance. One of the core concepts is the idea of driver mutations, where certain mutations are more likely to lead to further mutations hence "driving" to cancerous developments.

The infamous BRCA gene is one such example. If one person inherits a mutated copy and then develops a mutation in the other copy, they lose the capability to prevent damaged cells to undergo programmed cell death (apoptosis). When combined with other risk factors (female sex, radiation exposure, OCP use, etc.), random mutations can pile up that would normally be prevented by functioning BRCA gene products. Colon cancers also show a similar line of progression (APC > KRAS > p53 > VEGF genes) where the initial inherited mutation increases risk of developing cancers.

Another classic example would be the Li Fraumeni Syndrome, inherited dysfunctional p53 gene. The p53 protein is crucial to detecting damage to DNA and then initiating repair or cell death. Cells with one inherited mutated copy of this gene who further develop "non-hereditary" mutation in the other copy now lack this the capability for DNA repair in future cell divisions (two-hit hypothesis). Most of the time, other repair/immune mechanisms will prevent cells without a single functioning copy from turning cancerous; however, in cell lines that physiologically divide rapidly (skin cells, GI cells, bone marrow), cells can escape these mechanisms. Families with Li Fraumeni Syndrome can present with a variety of cancers in their medical history ranging from breast cancer to brain cancer.

There still seems to be a lack of literature espousing the benefits of using egg freezing to reduce negative outcomes. On the other hand, evidence indicates no difference in negative fetal outcomes of using egg freezing aside from cost/economic barrier. Risks to the mother of having an advanced age at gestation still remain but the technology is a boon to those who may be have decreased fertility later in life (ex: cancer, autoimmune disease, premature menopause).

With regards to aneuploidy (which includes Downs Syndrome), there is less risk by using oocytes harvested at an earlier age as opposed to later in life.

Great read looking over research from the past decade: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4467930/