Wouldn't these self-replicators also be bottlenecked by materials and other infrastructure? Does each one have a semiconductor fab, mining equipment, metal foundry, etc, built-in? I also don't see how you get from a laundry robot to a self-replicator through garage tinkering, those seem very far apart.
Does it set up a backyard forge and start to cast new arms and legs out of discarded cans? Does it finish the parts on a manual Bridgeport or does it carve them out manually with a hand file?
When it needs silicon chips, does it make them itself or try to order more from the company that made it? Will it become a right to repair activist when it finds out the company won't sell components to individuals?
Maybe it has to pay for all this with a part time job at the local fast food joint.
Assuming a humanoid robot with human like intelligence it may also be able to enlist humans to accomplish it's task, either by paying them directly, or through intermediaries, or deceiving them into helping it.
If it is capable to human like intelligence and creativity it may be able to pioneer new manufacturing processes -- perhaps it will learn to grow parts for itself by using existing biological processes in ways that we haven't yet figured out.
The point isn't so much the how with a self replicator, the point is the exponential growth rate. You're right that the first machine will take a long time to build the second, but those two will certainly be able to build twice as many in at least the same amount of time -- probably less because they can use the infrastructure that the first set up to build the second.
Once humans make machines that are capable of self replication regardless of where it is on the spectrum from base matter and energy assembly to just off the shelf parts assembly you're going to see exponential growth.
And once certain kinds of creative people get their hands on these machines they will inevitably jailbreak them and make them work for them instead of the companies that will try and lock this kind of stuff down like they always try to do.
The war on general purpose computing will transition into a war on general purpose manufacturing and the same kind of people who want ot sell you devices that you can't compile software for without a license will try to do the same with self replicating hardware but they will fail.
Robots that can build things already exist and can be purchased second hand, but I haven't see this level of hacking them to self replicate. The logistics of getting enough parts, and having the general purpose ability to diagnose a very wide range of issues with used components, and similar seem to prevent this from being some sort of self replicating singularity.
If such a robot did already have the power to do such self replication, why would it wait for a second hand hacker instead of the builder of the robots using them to produce copies at reduce cost?
Now imagine it's been a few years and one of these robots that used to go for the price of a nice car is outdated and can be bought for a couple grand and with a couple grand for a replaced battery and maybe upgraded hands for more dexterity.
Let's say an industrious young hacker gets their hands on this device and after fixing it up and jailbreaking it decided to get it to do stuff -- what's the first thing they should get it to do?
Why not see if it can assemble a copy of itself -- if you find a genie in a lamp why not ask it for more wishes?
The second a certain kind of mind gets their hands on a self-replicator is the second everything for humanity changes, the economy will never be the same because any task that used to be bottlenecked by materials or labor is now more or less bottlenecked only by the time it takes self replicators to build copies of themselves to divide and conquer the task.