When an IgG's Fab region binds an antigen its conformation changes leaving the IgG's Fc region more exposed for binding to the Fc receptors on phagocytes, or in this case platelets. The Fc receptor on platelets have particularly low affinity for IgG. Because of this, they are less likely to bind free floating IgG (with Fc region less exposed) and more likely to bind IgG bound to antigen.
The role of Fc receptor platelet activation in our immune system is not completely clear but yes, the platelet's Fc receptor is capable of binding other IgG-antigen complexes other then IgG-heparin-PF4. There's evidence of platelets participating in the clearance of microbial infections as IgG-antigen coated platelets are cleared in the spleen by macrophages with a corresponding Fc receptor. There also appears to be signaling cross talk between activated platelets and other immune cells.
The severe maladaptive activation of platelets in HIT happens because 1) heparin binds PF4 exposing immune epitopes leading to the formation of IgG, 2) the platelets Fc receptor binds the exposed Fc region of IgG-heparin-PF4, 3) platelet activation leads to even more PF4 release creating a positive feedback loop of 1 & 2. This process is also mediated by a) the binding of heparin to PF4 which is dependent on the molecular characteristics of the specific sample of heparin you are administering, b) the ability to form anti heparin-PF4 antibody which varies based on the patient population, the type of heparin used (greater risk with bovine unfractionated), and even the setting of heparin administration (more common in post surgical cardiac patients), and c) the binding of antibody-heparin-PF4 to platelets which is influenced by the titer and type of antibody created.
So yes, platelets can be activated through binding of other IgG-antigen to their Fc receptor. This response is just exaggerated in HIT with certain patients because of the circumstances.
Sources:
http://m.circ.ahajournals.org/content/111/20/2671.full
http://www.medscape.org/viewarticle/569661
http://onlinelibrary.wiley.com/doi/10.1111/jth.12905/abstract
http://www.ncbi.nlm.nih.gov/m/pubmed/17046568/
http://m.jimmunol.org/content/162/7/4311.full
https://en.m.wikipedia.org/wiki/Fc_receptor
Hope that helps!