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Tech support is a good grounding for admin, because it gives you a good handle on what needs to be administered. For instance resetting a users password and or rights is the business end of of having a strong password/security policy to set up and administer.
Similar examples can be made for any style of admin, apart from badges who happened to be the MDs nephew, most start in tech support. I did and I'm a developer.

I completely agree, any company that has a problem with you wanting to advance isn't worth the effort.