Self-drilling fine thread screws for hardwood

[Lynden]

I have 3/8" cherry pyramid plugs, but that limits me to screws with a head size of 3/8" or less. 

All I need is a good self-drilling screw (seems like most nowadays are) with fine threads, a length of about 2 1/2", and a head size (preferably washer head) of 3/8" or less. Is this a unicorn?

If you are going to plug the holes, couldn't you use a countersunk washer under the head of a 2.5" fine-thread drywall screw. The bugle head of the screw wouldn't fit the washer exactly, but that shouldn't matter. 

https://www.mcmaster.com/#standard-washers/=152emgf --> click on countersunk --> #4 OD = 0.375"

http://www.leevalley.com/US/hardware/Pag...1306,41316 --> #6 OD = 5/16"

[MichaelS]

Grex or McFeely both make world class screws, but no screw will self drill AND eject waste - which is what will result in the lateral pressures that will lead to a split. Go with a pilot, a split can occur as you drive the screw or months down the line and this will save you from the heartache.

I'm a HUGE fan of the Lee Valley kit with taper bits and countersinks, which together with Grex or McFeely eliminated my screw issues (I had 2 zinc screws break off and ruin projects which ruined my mood towards them forever...)

Michael