OP, if you go with putting the vinyl OVER the tile there are two things you must consider. first there will be a height difference wherever you had rugs. Second, (and I have personal experience here from a project up north), you may not be able to remove the dishwasher with the additional clearance (the height) of the vinyl flooring. If you have laminate counter tops, you might be able to break the glue and get the top high enough, of you have stone, forget it.
Bottom line, I completely agree with removing the whole floor, this is a relatively dusty and dirty job. You might do this in 2 steps, remove the carpet, and install vinyl in those rooms, live in the house for a while and see if you still need to remove the tile.
Hope this helps, and feel free to send a PM (private Message)if you want to discuss over the phone.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV.
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