Much of what you say makes no sense at all so reading betwen the lines:
By "thick" you mean diameter of holes through the counter,all sinks and counters have larger holes than nessary. Yours is normal.
By "gap" you mean the difference between where facuet and counter where facuet passes through hole,not to worry,it is normal and will not effect mounting facuet.
The manufacture has assured you it will work,it doesn't get better than that.

Standard drilling and sink castings are 1-3/8".
All sink fixture mfg. design and develop their fixtures to accommodate this standard.

Take, for instance, the 1/4" hot/cold water dispensers.

Thank you both! Yes, klem, you interpreted my layman's description correctly. Glad to know you both think it will work.