Learn English – say “thank you always”

The phrase in the title conveys what it says: an inability to rustle up more thanks. However, the connotation depends on context: words, smile/frown, situation, etc.

It could mean that the little appreciation you’ve expressed is all you can summon, or that you’re so filled with gratitude that you feel like you’d float if you were any more thankful.

In your context, you are expressing sincere gratitude with the phrase, especially if the rest of what you communicate says likewise.

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.