I'm self-taught in most fields that pertain to programming, software engineering, computer science, and electronic engineering. I have interests that range from 3D Graphics Programming, 3D Game Engine design, hardware emulation, compiler design, os design, hardware design down to the circuitry and from my research and studies I think the best way to answer this is that it comes down to how hardware is actually built and its constraints.
In mathematics, we have notations of values that exceed the limitations of physical hardware. Random Access Memory is used in many computer models, however, we have to look at the nature of what a Turing Machine is and how we use it to perform mathematical computations. We can represent integers and reals, but there are always the underlying issues of precision, truncation, under and overflow of values.
You had stated:
Turing machines are perhaps the most popular model of computation for theoretical computer science. Turing machines don't have random access memory, since we can only do a read where the slider is currently located.
This seem unwieldy to me.
Then you had asked:
Why don't theoretical computer scientists use a model with random access memory, like a register machine, as the basic model of computation?
We need to look at the definition of what a Turing Machine is:
A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of simulating that algorithm's logic can be constructed.
provided by: wikipedia
Do yourself a favor and read over the wiki page thoroughly and consult other resources as this has widely been used as the de facto model for most modern computers.
To answer your question you should read the details from the wiki page about how a CPU acts like a Turing Machine and how most programming languages are Turing Complete. It comes down to the layers of abstraction which is in contrast to a what a Turing Machine is while being an Abstract model.
In a computer layout design typically the CPU the Turning Machine has a finite state of registers with finite register sizes and bus widths along with a finite set of instructions that can be performed. The dynamic random access memory modules are normally connected to but separated from the CPU. Internally and at each abstraction layer, a computational device is a finite state machine independently but when combined to build a more complex machine, theoretically can perform an infinite amount of instructions.
If you were to use either a breadboard with integrated circuits, an FPGA, or even a simulator program such as Logisim and start building a basic CPU with its data paths, buses, control lines, etc. You will end up seeing the limitations of the hardware when you start to see the connections between the design of the logic gates, the component layouts, and how an ISA (Instruction Set Architecture) is designed to produce a Turing Machine.
When you begin to understand how an Assembly Language is constructed from an ISA and is used to simplify the ability to talk to the machine in its machine language you will then be able to build higher levels and layers of abstractions on top of it such as the C family of languages, Compilers, Operating Systems and such. Then you will end up with more advanced topics such as compiler theory and Operating System designs.
It pretty much comes down to two main things, finite state machines acting as Turing Machines and the layers of Abstraction. There are two viewpoints that this can be seen from, bottom-up and top-down. Looking from the hardware up to the software or looking down from the software to the hardware and understanding how it works where they meet in the middle. This is where most of the theory is applied. In the end, it is basically a hierarchical design where it is one system built on top of another.
A Turing Machine in itself doesn't require random access memory, however, many Turing Machines are connected to and work with random access models in tandem. They are two separate things that I believe you are trying to compare... Apples and Oranges, yes they are both fruit, but far from the same. Yet, we can use both of them together to make a fruit salad!
Basically a CPU is a Turing Machine that has Registers with a finite set of states, but in a computer system, the CPU is connected to Memory Modules that are Random Access through the system bus or a bus controller. Without the two you wouldn't have a computer system. They talk and communicate with each other to perform a set of instructions based on the current or previous instruction that has already been executed, in memory ready to be executed, and so on...