Before I get into the Answer proper I wanted to say that I'm presuming from the outset that you don't need to produce perfect blanks like many commercial ones are. That level of finish is useful for marketing and sales because it gives a clearer picture to prospective buyers of exactly what they're getting, but you don't need to aim for that since you have the wood right in front of you.
And working towards getting surfaces that nice, when none of them will remain on the finished item, is pointless and you'll save a lot of time and effort if you produce blanks that are just good enough to work with.
There's tons of post-assembly shaping involved in knife handles so I'm assuming it's similar for razors, so you really only need the pieces to be flat (no bow etc.), of even thickness even if only approximately 1/8" thick, and reasonably smooth (a flawless surface is actually not desirable if bonding with epoxy and using light clamp pressure).
So how might you do this? Instead of concentrating on the actually tools you'd use to do the various steps here's the approach I'd take, which can be done in various ways:
- Buy stock in board form.
- Resaw over-thick to allow for flattening and getting to rough thickness.
- IF NEEDED plane one face to get the wood to thickness. The other face doesn't matter unless you have OCD and can't bring yourself to work with wood that has a rough surface.
- Rip strips of suitable width.
- Cross-cut strips to length.
The wood's ready to use then.
Obviously you're going to quickly generate enough blanks that you'll need to store some since only one longish board 1/2" thick is going to produce more than you can immediately make use of, so it might be important (depending on the wood, your local conditions and any expected changes to those) to store the stock wrapped in plastic wrap or inside airtight plastic boxes to keep the wood stable.
Now, which tools to do the above?
You could do this using hand tools. This might sound crazy if you don't use hand tools much but every process is perfectly doable by manual methods, especially if you set up a production line to do repeat steps in one go.
This is the way I'd do it since I own no power saws, but I'm not suggesting it for that reason. Since you don't currently own the tools necessary to do all the steps (at all, much less safely) you have to do some buying and this is by far the most cost-effective route, although obviously there will be some sweat equity needed as well.
If you want to buy in a power tool to help with this, then I highly recommend getting a bandsaw. Bandsaws are the power saw for doing resawing and can rip and crosscut effectively too, so fairly easy to set one up to do all the sawing steps you need here, and down the line a bandsaw will give you the most bang for the buck because of how versatile they are.
Additional safety note:
a very old and craftsman table saw on a wobbly stand with no guard.
If it also has no riving knife then I would strongly strongly recommend you never use this to cut anything. Until such time that you've sorted out at least one of the important safety features (ideally both!) and have fixed the stand it's an accident waiting to happen, and accidents at the table saw can be life-changing or life-ending events.