I have switched several machine tools to 3-phase with VFD run off American single-phase 220V or 110V (depending on HP) or to DC with a speed controller (KB brand) runoff 110VAC.
It is a bit of a toss up. VFD's have a definite lag before starting. DC starts and stops almost instantly. Since your mill is bench top, I suspect it is less than 2 HP, perhaps even less than 1 HP.
I would not mess with trying to replace the 600V, 3-phase supply, unless it has a purpose-made motor and another motor (e.g., a flange mount) will not fit. If it is less than 1 HP, I would lean toward a DC conversion, if you can find a motor for a reasonable price with the right mount. Price wise, a 3-phase motor with VFD will probably cost less, particularly if the motor is not fractional HP.
Another factor to consider is what type of speed control does the mill currently have. If it is done by changing belts, I wound be even more resolved to go to a variable speed drive. Changing belts gets awfully old on a mill.
Does your mill have power feeds (e.g., Servo feeds). If it does, be sure to consider what line voltage they need. If it doesn't, you will probably be adding power feeds and DRO. Trying to stay with the 600V head will mean bringing different supplies for those additions.
John