The smaller the p-value, the higher the likelihood ratio: true or false?

[Thanks to Scott Glover for correcting me.] Someone recently said to me that the lower the p-value, the higher the likelihood ratio under the alternative vs the null. The arXiv paper by Michael Lew makes analogous points (thanks to Titus von der Malsburg for pointing me to this paper).
I show below why this question is irrelevant. The problem lies again with Type M errors under low power. The bottom line is: should I care about a significant result if it is based on an overestimate of the true mean, i.e., if it is based on a biased estimate of the true mean? PS I corrected a typo, thanks to Keith Goldfeld.

Here is the blog post.