how many screws are needed for 300 lbs
Let's make a few assumptions to be clear about the situation. 


A 300 lb. person stepping down at a conservative maximum of 2 g's (600 lbs. of force)
Load split between at least two supporting members
No more than 80% load on any one member
No additional load from other persons


This means that any one member should be able to handle a load of 480 lbs. plus a safety factor. 

In my experience* with SPF lumber, screws of fair quality pull out before they break, except when experiencing repeated bending stress due to displacement of the joined members. This means that in the case of a well-fastened framing member, no significant movement should occur. 

In my estimation, a member fastened into SPF lumber using four 3" screws (where penetration into the supporting member is at least 1-1/2"), and which is supported against rotation through being fastened to an overlaying member, will carry around 500 lbs. without undue risk of failure. 

This is too close for comfort with your requirements. I'd go to six screws, recognizing that this may be challenging with your stair cleats. I'd also look for a heavier screw, say in a #12--pullout strength increases exponentially with screw diameter/thread area. Piloting to prevent cracks is key, and vertical staggering is also important.

So for our 450 lb. load, 12 screws or more would be adequate to support each plane. Again, it's not the strength of the screw that's our weak link, but pullout in fairly soft lumber. With that in mind, consider going with three or four 5/16" or 3/8" lag screws per member instead. You'll need fewer and have more peace of mind. 



* As a guy who's built a whole lot of stuff over the last four decades but done little engineering calculation