I've been dealing with a problem where standard screws keep getting pulled out of a wood frame.
Even with options on screws I think the first thing you should look at is why are your current screws pulling free? It sounds like there's an issue with one or more of the following:
A) the wood being too weak (in which case no screw of reasonable gauge may hold successfully)
B) the load is too high for the number of fasteners (if possible increase the number to spread the load)
C) the pilot holes were drilled too large
A and B can be linked, so a fix for A can mean that the existing number of screws will be sufficient now that they have a firmer hold.
Regardless of whether C was the case initially since you've already had screws pull free I think we can assume you have stripped holes, which means you have to go up a gauge or two in screw size and/or repair the existing holes before driving in new fasteners (see Loose screws in this previous Answer).
If your wood doesn't have the strength to support conventional wood screws then one of the fixes for a stripped hole can be used to provide a more secure hold, including drilling out for a glued-in hardwood dowel or wooden plug (note: only drive screws home after the glue has fully cured). In addition to inserting dowels or plugs where the screws will go dowels can also be installed at 90° to the screw's axis. When driven home the screw then bites into the side of the dowel (long grain) which provides a very much stronger hold if it is going into end grain. This is often done if strength is important:
Are there such things as screws that have "oversized" threadform? (if so, what are they called?)
Coach screws can have larger threading in comparison with a typical wood screw, although this does tend to go hand in hand with them being quite beefy. Many would to too large or ugly for some applications. Although there is some variation they typically look like this:
Many modern deck screws also have larger threading (or twinned threads).
Most threads seem to be between .60-.75 pitch, but is there something on the scale of 1.5 pitch?
Machine screws have a lower pitch. Although they also have a very shallow threading usually if they fit their hole well they can have an extremely good hold in wood (hardwood in particular). Two reasons they're not more widely used is that they're slow to drive and the head style is usually not what we're looking for.
Or would the material of the thread not be strong enough to add any practical gripping power?
The wood is nearly always going to be the weak point here. It's the threading in the wood that may not be strong enough as it's created partially or wholly by crushing of fibres. There's little fear of the thread on the screw itself stripping off unless it's been damaged, e.g. by corrosion. But even at that I've pulled heavily rusted screws and bolts from wood numerous times and the threading usually remained intact!
