plotting - RevolutionPlot3D: axis of revolution that doesn't pass through the origin

This is a follow up question to RevolutionPlot3D: but NOT revolving about the z axis so please check that for the context.

It seems that RevolutionAxis requires the axis of revolution to pass thru the origin. Suppose I want to use an axis of revolution that does not pass through the origin, e.g., the line $y=-1$ in the example above. What is a good way to accomplish this?

I do realize that RegionPlot3D may be appropriate here, but even with seemingly simple examples such as the one above, it can struggle:

Show[
  RegionPlot3D[1 <= Sqrt[(y + 1)^2 + z^2] <= 1 + x^2, {x, 0, 1}, 
    {y, -3, 3}, {z, -3, 3}, PlotRange -> All, PerformanceGoal -> "Quality", 
    Mesh -> False, AxesLabel -> {x, y, z}, PlotPoints -> 100, 

    ViewPoint -> {0.87, 0.44, 1.76}, ViewVertical -> {0.32, 0.67, 0.67}],
  Graphics3D[{Thickness[.01], Black, Line[{{0, -1, 0}, {1.15, -1, 0}}]}]
]

enter image description here

Bumping PlotPoints up to 200, 300, 400 doesn't alleviate the problem and gets really slow.

Comments

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