If we had infinite memory and infinite symbols (instead of only 2^64), we could build a universal Turing Machine. But that doesn’t mean our physical computers aren’t Turing Machines executing Turing-complete programs that just happen to be in the space of programs executable in a finite time and within the limitations of our current physical realities.
I’m not sure who told you that Turing never intended for the machine to be built. He knew that he could not build a machine with infinite tape, as we both know too. But when he built the machine to crack the Enigma Machine, he did this _after_ writing his paper, “On Computable Numbers.” So, he was in effect making a Turing Machine (but he called them a-machines).
We literally didn’t have computers before Turing formalized a way to execute them, instruction by instruction, and in detail. Once we had some kind of formal method to do this, physical machines could start being built.
Edit: My comments have not shown misunderstanding. There seems to be this consensus that our mathematics here came _after_ our machines. But this is wrong and the implications are tremendous. Our computers we have really are machines which execute other machines. What is the only class of machine that can do that? The Turing Machine.
Edit2: I suppose this is an argument of semantics. I didn’t realize. Yes, perhaps our computers are “Turing-equivalent”. I still think the “tape” and “symbol” and all of those things are merely mental variables, which is why I called these things a Form earlier. We built a physical tape, it just took the form of RAM and has much detail that electrical engineers can talk about. We also built symbols, we just called them instructions and placed them into a CPU. All of our physical devices have come from theoretical findings, not vice-versa.