Platelets are a specific cell type in normal blood. They don't carry oxygen, so they don't provide the normal advantage of "blood doping". Moreover, the re-injection is into tissue, not the blood system. The (claimed) advantage of platelet injections is thought to derive from their special ability to secrete a variety of "growth factors", such as PDGF, FGF, TGF-beta, which are normal protein components of blood that regulate the growth and development of many types of cells. Injecting a large bolus of platelets into an area of injured tissue, the logic goes, is a way of promoting the natural healing process with a wide spectrum of the body's own healing agents. It is legitimate medical practice to promote healing by activating these pathways.
The gains should be short term, since platelets (which don't have a nucleus) live only a week or less, and local (since the platelet-derived factors are used and degraded mostly where they are produced, at the injection site).
However, it should also be acknowledged that Epo (erythropoetin) is also a growth factor, although specific for producing red blood cells (erythrocytes), which is normally present (like the platelet factors) at moderate levels in blood. Epo is a hugely beneficial factor for cancer patients, whose ability to survive their drug regimens depends on replenishing the blood cells killed by the drugs (which target dividing cells). However, elevating Epo levels in normal persons produces unfair benefits in muscular performance, specifically by increasing blood oxygen-carrying capacity.
Similarly, it is definitely possible that a regular program of platelet injections, or cocktails of platelet-derived factors, could boost performance, as for example by boosting recovery during heavy training (which is/was a common motivation of ballplayers using steroids, indeed McGwire's open justification for using the now-banned steroid precursor androstenedione).
Teg cannot be justifiably doubted for his sub13 run, and medicinal use of local platelet injections should continue to be accepted practice for injury recovery in anyone, athletes included. But i'd argue that there also should be a pro-active control system put in place to intercept possible abuse of systemic platelet boosting, at least until evidence showing a lack of efficacy for performance is demonstrated.