Hey Bruce!
All good observations above. While I am not an Ithaca pump gun authority, I have handled and fielded many of them in the past. In my experience, the game scene engraving or lack of it appears intermittently in all of the commercial models of the Model 37. As I recall, there is more than one pattern of this engraving with the later type an acid etched, relief type similar to that of Remington's in the mid-90's on their 7400, 7600, Model 4 and Model 6. Of the two patterns, I think that the earlier type as seen in Tom's excellent photo to be the more tasteful. As I recall, the 20 gauge field model I just gave away also had this engraving. However, the folding stock featherweight police model I once owned did not.
Either way, the 7-shot Police Special you discussed would make a fine addition to your "Father and Son" battery, especially given the fact that you already own one. About the only thing I would do is change out the magazine tube spring just to be sure it's fresh.
$300 is a very good price in my view, especially if the gun has rifle sights and the correct recoil pad. A Good Remington 870 Police set up the same way would cost you this amount and more....