Why haven't any teams tried to use the Two-Way Player rule?

Over the winter there were some articles suggesting Michael Lorenzen was marketing himself as a TWP in FA. Now this is probably a stupid question, but why haven't any teams tried to utilise the TWP rule to carry an extra pitcher on their active roster?

To qualify as a TWP you need:

• Pitch 20 innings
• Start 20 games as a position player, with at least 3 PA's. (can be DH)

Now, you could probably say giving a pitcher 3 PA's for 20 games gives less of a chance to win a game. Sure, in the short term. But what about the long term benefits of carrying an extra pitcher on the 26-man for the next 2 years? Obviously teams would have to believe its worth having an extra arm over an extra bat or pinch runner. But before the 13 pitcher limit implementation in 2022, 21/30 teams used 14 pitchers on their 26 man roster.

I guess it would come down sacrificing 60 PA's for a pitcher over a 2 year period. Michael Lorenzen has a career BA of .233 across 147 PA's. Max Fried has a BA of .273 over 67 PA's. Albeit these where a long time ago, but I'm sure there's some pitchers rocking about that are at least decent with the bat.

Would it be worth even obtaining the status just as roster flexibility? Recent changes over the winter meant that you keep the status over 2 years, even if you fail to maintain the requirements.

Just a random though I had.