I know this against code and unsafe but can someone explain why I shouldn’t and what the potential scenario is that would lead to the unsafe situation?
There are a lot of possible scenarios, and the biggest reason it's hard to get a straight answer to that, is without an accurate and complete schematic diagram of your entire electrical installation from the meter onwards, it's hard to tell you which failure modes are likely, possible, or even conceivable.
The idea of the dedicated ground (the "equipment ground") is that no current whatsoever is flowing on it, ever. If you allow current to flow on it, there is always some voltage drop. (Wires themselves have resistance, and drop some voltage). You will end up with, somewhere, potential differences between exposed items, including any panels you have installed. All the bare wires bonding anything in the house are non longer waiting there to carry current for a millisecond, in order to trip a breaker, but they are conducting current. What could that do? Which house do you live in? There could be a different set of answers for every member of this forum.
It's not so much that some specific thing could or could not happen to you. The real world reason this should not be done, is that you do not know what might or might not be at risk, or how much risk. It could be nothing. It could be everything.
I have 4 wire single phase nearby but the lathe came with a 3 wire whip on it. I need to get an outlet for it, I could change the plug style and whip but that costs a little more.
There's your answer right there..... You need to get something..... Stew on this for a minute... It still cheats a bunch, but it doesn't affect the the rest of the house, only the lathe. You have four wire single phase available near the machine. Leave the building wire intact, with the correct four pole receptacle which is either already there, or you need to identify and source the correct one. Wire that receptacle absolutely correctly. Hot, Hot, Neutral, and Ground. Then, on the plug end of the whip, install the matching four pole plug. Depending on what cord the lathe came with, you might need white tape to reidentify the ground wire in that cord as neutral. Do that at both ends, even if it clearly just screws to an electrical box or motor housing or whatever else, reidentify it. That's the reminder that it's no longer a "ground", but a "neutral".. Attach that re identified, now neutral wire to the Neutral pin in the plug.
By solving the problem "outside" of the receptacle, instead of the building wire, you will not affect the design intent of the building's wiring, no current going through the things that are bonded to ground, not put current parts of your building that were designed to last ONLY long enough to trip a breaker, you will simply make your building as it should be (or at least how it is now), and the lathe as it would have been before enough people died that the dedicated equipment ground became mandatory. Prior to the equipment ground being separated and manditory, this is how it would have been factory wired if it were designed to have 120 volt accessories. (Or 117 volt, or 115 volt, or 110 volt....).
Still not as good as doing it right, but the compromise is well documented, well known, and very isolated to the lathe it's self. The rest of the house is exactly as it was, no new risk, no new compromise, nothing about this affects anything, anywhere, except the lathe it's self.