It is in fact extremely common to build wholly deterministic machines out of the same parts as our unreliable computers, that are correct by construction. And they may implement a Turing-complete language and still be wholly deterministic, just by limiting the input to what can be proven deterministic in a strictly limited time. Again, this is normally by construction, where the proof is always trivial.
Such a proven-correct system can model literally any space-constrained algorithm.
It is neither common nor easy but actually provably impossible. Please read this piece carefully: https://pron.github.io/posts/correctness-and-complexity
You must be misunderstanding the work if you believe that what is being done routinely is impossible.
Quite possibly I don't understand what you mean by "work". If by "work" you mean creating provably correct software then no, it is not done routinely.
People create wholly deterministic state machines all the damn time.
What do you mean by "wholly deterministic"?
You can buy an 8-bit adder/accumulator from a catalog.
We still don't know if it is possible to build a provably correct multi-tasking OS or a web browser without security vulnerabilities. If it was easy we would already have one written in FStar or Idris.
While admiring all the work done by Project Everest - the fact is that it is a multi-year project (I first read about it about 3-4 years ago and it was already under way) with a very limited scope - it is more of an evidence of difficulties of writing correct software.
Translated: there exists a computational model weaker than TM but strong enough to implement programs we care about.
Unfortunately, we don't know such computational model and we know empirically surprisingly many problems require a TM (ie. surprisingly many languages are accidentally Turing complete)