Both are acceptable. "I would like to thank" is slightly more polite, because it is a kind of "indirect" speech.

Grammarly flags this because (I assume) it is built on a particular style guide that emphasizes direct speech patterns. With expository writing, qualifications might sound "weak". For example:

I think it may be true that the African Elephant is larger than the Asian Elephant.

Instead you should know whether or not it is true, and say so with confidence:

The African Elephant is larger than the Asian Elephant.

However confident, direct speech is not always the best option in every situation, especially when expressing humility and gratitude.

I would like to thank my parents, without whom I wouldn't be.