Let me assume that you are concerned with ordinal time Turing machines, using a tape of order type Ord. 

My first observation is that the accidentally writable reals are exactly the constructible reals.

**Theorem.** The OTM accidentally writable reals are exactly the constructible reals, that is, the reals in $\mathbb{R}^L$. 

**Proof.** The forward inclusion holds because all the OTM computations can be undertaken inside $L$, and so whatever it is that appears on the tape at any moment for them will necessarily be in $L$. So every accidentally writable real is in $L$. 

Conversely, we use the fact that the machines are able to simulate the constructibility hierarchy. With a suitable choice of finitely many ordinal parameters, the machines can construct a code for any desired level of the $L_\alpha$ hierarchy and pick out the code for any particular constructible set. In particular, with suitable parameters, one can produce any given constructible real on the tape. And now the point is that we can design a program that systematically does this for all possible choice of ordinal parameters. The universal algorithm will simply iteratively increase a master ordinal, interpreting it as a code for a finite tuple of ordinals, and carry out the construction that far. So every particular constructible real will appear on the tape during this universal procedure. $\Box$

In particular, the supremum of the OTM accidentally writable reals will be exactly $\omega_1^L$. 

Meanwhile, there are only countably-in-$L$ many programs and therefore only countably many eventually writable reals, since each one can be associated with the program giving rise to it. So $\eta<\omega_1^L$.

If I understand the thrust of the rest of your question, you are proposing to design and algorithm that will try to stabilize on $\eta$, which of course would violate the definition of $\eta$. I believe that the answer is that you are not paying attention to the fact that we are talking about real codes for ordinals, rather than ordinals. At limit stages, we perform the limit operation of the machines, which is not the same as taking the supremum of the ordinals coded by the reals. So when we try to stabilize at $\eta$, what happens instead is that at certain limit stages where we become unbounded in $\eta$, the limit of the codes is not coding $\eta$ but some much smaller ordinal or is not even coding an ordinal at all. 

(Some of your remarks suggest that you are thinking of a register machine model, instead of the tape model, which you mention in part of your post, and so I am not actually sure what you have in mind with some of your remarks.)