First of all, it's very good to think about creative ways to build a splitter and save $$. A threaded rod solution is feasible and in the early days of manufacturing there were many threaded presses and press type tools in service. Compared to hydraulics there are specific problems that you will need to consider.
First, the threads of the rod and whatever you use for a nut are going to be subject to force and friction. If you are going to apply 2000 lbs of force to the log then your threads have to be able to withstand 2000 lbs. Repeated use, back and forth, with pressure included will require a high quality extreme pressure lubricant on the threads to prevent siezing and galling. Oil rather than grease based lubricants are preferred. In the shop I have found that cleaning the threads of my ball joint press after each use and applying clean 90 weight gear oil provides good, long-lasting results.
Second, torque applied through the rod when splitting will tend to twist or bend the rod. A bearing at each end is ok for a shaft that is not under extreme torque but for high pressure it will be important to ensure the pusher block doubles as a guide to prevent the rod from bending when splitting load is high.
Third, the rod will be subject to both stretching and compressing which will actually cause it to change length. You should make sure one end is mounted in a way that will allow some change in length. Additionally, if the beam of your splitter deflects under a tough log then your drive screw mounting should allow for this if possible.
Fourth, assume some logs simply cannot be split and build accordingly. If you build a drive system for the threaded rod which has no clutch and no pressure relief, what will happen when your unstoppable force meets an immovable object? A pressure relief valve is easy to install in a hydraulic system but with a mechanical system you should have some type of clutch, preferably automatic, to allow the prime mover to be disconnected from the splitter. I have a funny story from my youth about a boy who powered a bicycle with a chainsaw engine but forgot to include a clutch.
Fifth, consider designing the splitter to both pull and push at a reasonable speed. A hydraulic ram's speed can be controlled by the volume of fluid allowed through the control valve. A simple, single speed drive system for a screw-type drive might provide reasonable splitting speed for a knotty piece of oak but when you hit nice, straight grain wood you might want to run a little faster. And when it's time to retract the pusher and put in another log, moving that slow speed pusing block will become the most frustrating experience ever.
Some other advice is to check the thread type on the threaded rod you're looking at. Some threads are better at carrying pressure than others. A typical thread on most fasteners is made of 60 degree angles and tapers to a point. The thread is strong at the base but gets significantly weaker near the tip. At one time square threads were used for high pressure but in most cases today, Acme threads are preferred due to ease of machining. Acme threads have less angle than UNC or UNF threads and are typically machined to tighter tolerances. They are commonly used when more pressure is expected and they're a first choice for pusher bolts in the US. Typical places to find Acme thread is on the bolt that moves a vice jaw, on the pusher bolt of a threaded a ball joint press, and on the bolt on a scissor jack.
Typical bolt thread:
Acme thread:
For price comparison, McMaster Carr would sell you a 1-1/4"-4 X 6' Acme ASTM A193 threaded lead screw for $216 and the matching mild steel coupling nuts could be had for $44 each.
