Why is cisplatin a very potent antineoplastic for testicular cancer, but not necessarily for other cancers?
From the dedicated cisplatin website and Drugbank it appears its use is quite diverse and not confined to testicular cancers. Further, your linked (Siddick, 2003) paper mentions on the first line of its abstract that cisplatin has clinical activity against a wide variety of solid tumors.

Cisplatin.org mentions that cisplatin finds use as chemotherapy against a range of cancers:


  Cisplatin is a chemotherapy drug which is used to treat cancers including: sarcoma, small cell lung cancer, germ cell tumors, lymphoma, and ovarian cancer. [I]t can be a valuable part of a combination chemotherapy regimen. Look at the regimens given to patients and you will often see cisplatin as one of the drugs. 


Moreover, cisplatin is explicitly used to treat ovarian cancer:


  Treatment of ovarian cancer involves putting the cisplatin into the peritoneal cavity rather than a blood vessel. The type and extent of a cancer determines the exact dose and schedule of administering the drug. 


As to your comment that cisplatin has "magical powers" I can say that it does not. The article that your linked cisplatin wikipedia page refers to (Einhorn, 1990) is a review paper that describes the efficacy of  combination therapies with cisplatin. The high cure rate of 75% was reached when cisplatin was used in combination with etoposide plus bleomycin. The last 10% was theorized to be achievable with 'salvage chemotherapy'. Hence, it is not cisplatin on its own. 

In general, it is common practice to use various chemotherapeutic agents in consort, preferably with differing modes of action to reduce the dose of each and hence lessening the side effects. 

References
- Einhorn, JCO (1990); 8(11): 1777-81
- Siddick, Oncogene (2003); 22: 7265–79