Is there a good reason NOT to reuse the power cord when replacing my electric range/oven?

As long as nothing is burned or melted on the plug end, and provided it is appropriate for the application (amperage wise) you should be fine reusing the old cord.

Personally... New! Especially as you know it is at least 40 years old. Irrespective of how well things are looked after they degrade over time and I don't think I'd sleep well at night knowing there is a potential issue just waiting for the most inopportune moment to rear it's ugly head.

No idea what your paying for the new range but I would think another $20 for a new cord would be a safe investment.

Just be sure to check your existing receptacle. Based on your stated age of you old range it is likely you only have a receptacle with 3 prongs. Two hots and a neutral but no ground. All codes and manufacturers now require grounds.

Be sure to read your warranty as it's possible the manufacturer's warranty will be voided if not properly grounded.

This will require not only that new $20 cord but new receptacle and 3 wire with ground from the receptacle to your breaker panel.

A big reason: the old cord (and receptacle) may be obsolete and dangerous.

Your connectors will look slightly different, since they are probably -50 instead of -30, that number being the amp rating.

40 years ago, grounding was relatively new, and it was "par for the course" to fake up ground protection for stoves and dryers by using the neutral instead. Most existing houses didn't have grounds, just 3 wires, and forcing everyone to rewire would kill stove and dryer sales - not economically (politically) possible. The rationale was that neutral connects to ground in the main panel, and dryers and stoves are rarely unplugged.

The problem is, if that neutral connection is severed for any reason, the 120V loads in the stove (i.e. the oven light) will lift the neutral to 120V. The user will get shocked from a 120V leg, through the clock mechanism or bulb filament, through the user, to the sink or whatever they may be touching. People have died of this.

For many years, NEMA 14 has been mandatory in new work. NEMA 10 is still legal if your home already has one, to allow new stoves and dryers to be sold into older homes... obviously safety cannot be allowed to slow down appliance sales. Some cities override this.

Every new appliance is built to accept a NEMA 14 cord, or if necessary a NEMA 10. This is why they want to sell you a cord: they don't know which you need.

If it's NEMA 14, you're all set. If it's NEMA 10, it's worth popping the receptacle cover off and seeing if there are four wires. If there are, change to NEMA 14. Otherwise consider rewiring - it's legal to add only the ground wire, routed any practical route - it doesn't have to stay with the conductors.