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It looks like you've been the victim of some misinformation. As Dr_Birdbrain said, an SVM is a convex problem with unique global optimum. sklearn.SVC relies on libsvm which initializes the weights to 0 [0]. The random state is only used to shuffle the data to make probability estimates with Platt scaling [1]. Of the random_state parameter, the sklearn documentation for SVC [2] says
Controls the pseudo random number generation for shuffling the data for probability estimates. Ignored when probability is False. Pass an int for reproducible output across multiple function calls. See Glossary.
[0] https://github.com/scikit-learn/scikit-learn/blob/2a2772a87b...
[1] https://en.wikipedia.org/wiki/Platt_scaling
[2] https://scikit-learn.org/stable/modules/generated/sklearn.sv...
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