No, visa-waiver applicants do not get visas at the border. The whole point of the visa-WAIVER program is that you do _not_ need a visa. You get a stamp showing your _status_, not a visa. When entering the U.S.A. you also get an I-94W if you are traveling under the visa-waiver program and an I-94 if you are visa-exempt (Canadian or Bermudan).
Again, the I-94(W) or the stamp in your passport IS NOT A VISA, visa-waiver/visa-exempt travelers do not normally need a visa, and there are NEVER visas issued at the border.
You see an I-94(W) or stamp in your passort, and perhaps think "Oh! A visa!" But it is not a visa.
Do me a favor: write to the U.S. gov't and ask them if they issue visas at their border at all. They will definitely say no, if you need a visa you need to get it from a diplomatic post, and if you are visa-waiver/visa-exempt you will not need nor get a visa under normal circumstances.
I hope I have managed to explain this properly to you this time, commonwealth, but I'm starting to lose hope...
The exemption to the six-year requirement applies if your application has been pending for over a year before you get into the status of an applicant for adjustment of status. This is because, as you mentioned, it is a multi-step process: labor market opinion, application processing, and then adjustment of status (or processing at the diplomatic post if applying from abroad).
Either way, H1-B is not a path to permanent residency, and you still have to meet all the qualification requirements for an immigrant category if you want to get an employment-based green card. Being or having been H1-B does not by itself give you a path to PR.