@GregoryLeo another side-thought is gearing changes.
Some assumptions - the bike is a 48/38/28 triple, with an 11-32 cassette (number of gears is irrelevant here) and I've assumed a 559-54 tyre, so ~54mm wide.
48 38 28 tooth chainring
11 113.2 89.6 66.0
12 103.8 82.1 60.5
14 88.9 70.4 51.9
16 77.8 61.6 45.4
18 69.2 ...
An old MTB will have used 26" wheels (559mm rims), you want to put 700c (662mm) rims on it with smaller tires.
We can work out wheel approximate diameters. Assuming tire height above the rim is about the same as its nominal width and the MTB frame will take a 50mm tire.
559/2 + 50 = 329mm
622/2 + 28 = 339mm (+10mm)
The 700c tire might fit but bear in ...
I think you'll find that most mountain and road bicycle wheels with mounted tires have the same outer radius (approximately). A road bike has 622-28 tires, for an outer diameter of 678mm. A mountain bike has something like 559-60 tires, for an outer diameter of 679mm.
The rim size is smaller on the mountain bike only to allow mounting a larger tire size ...
is it a MTB frame for a 29" wheel originally? then "probably" But if its a smaller wheel size then your chance of success will drop.
You also need to check the OLD of your frame vs your proposed wheels - a road bike wheel is probably 100mm at the front and 135mm at the back, your frame could be one of the weirder MTB sizes depending on its ...