[Manuka Jock]

You bought that table ? How much did you pay for it .

 

[Manuka Jock]

Splinters are bunches of loose (shattered) wood fibers . They cannot be smoothed down without first 'gluing' them down .
Try flooding the surface with a 50/50 mix of polyurethane and thinners , let it soak in and then flood it some more . Allow it to dry rock solid and then sand and fill and finish to your satisfaction .

 

[Manuka Jock]

I suggested seating the nails with a nail set. In using a random orbit sander (ROS), sand the table down to eliminate the loose wood sections. You can't "glue" the loose pieces down with a thinned polyurethane, as that would preclude doing any staining once dry.

Once the table is "surfaced", stain with an oil base stain, and then you can use a wipe on oil base finish like Watco". Or use a Watco that has a color you like.

He didn't mention wanting to stain the thing did he ?

Laying on the thinned poly will achieve the desired result

Maybe some kind of thin satin poly or something to cover the whole surface which would fill in the rough parts but not create a thick new layer.