This study link.springer.com/article/1... finds that taking levothyroxine with a sip of water reduces the amount of levothyroxine needed.
I believe the recommendation to take levothyroxine with a full glass of water is to avoid the tablet becoming stuck in the oesophagus. I don't know of any studies that were carried out prior to this recommendation.
We know that an acid environment in the stomach is needed for good levothyroxine absorption and that an empty stomach is recommended. When empty the stomach is contracted with a small volume. It seems logical that if you introduce a full glass of water it will substantially reduce the acidity of the stomach. This study seems to confirm this hypothesis.
The study has a number of pitfalls. The sujects were on a lowish dose of levothyroxine, averaging 69.1 mcg. Also, it was not placebo controlled, improvements over time may just happen. They could have done a simple crossover trial.
So it does seem that the recommendation to take a full glass of water is silly, we need a better trial to be sure. I take mine with orange juice but I've never had an absorption problem.
Most people are getting on just fine, at least in terms of absorbing their levothyroxine. In which case don't worry, be happy. If you are one of the minority who have absorption problems it might be worthwhile trying a sip of water and see how it goes.