Here is a raster-based solution for you.
On your PC, open the Microsoft Store and search for the "ProRaster Scientific" application.
You can install the app from the Store as a 7-day free trial. After 7 days it stops working unless you go back to the Store and purchase it.
Run the app and drop down the "Processing" menu button.
Select the "Classify (Polygons)" operation.
This will convert a MapInfo table containing polygons into a raster.
As a minimum, browse to the MapInfo table file and specify the output raster file and cell size.
After processing, the raster will be displayed.
Now, open the "Raster Source Editor" (there is a button on the app) and select the raster.
You can now display information about the raster including statistics.
Select the raster, drop down the "Info" button, and select "Detailed + Statistics".
Then, hit the Info button and it will display all the information.
In the report, look for "Spatial statistics for field 0 band 0".
Under this is the "Valid sample count:".
This is the number of populated cells that are inside the polygons.
Multiply the valid sample count by the area of a cell, and you have your answer.
We are "pixellating" the polygons, so the area will be slightly different to the true mathematical area.
You will want your original polygon table to be in a suitable projection.