In this day and age, does writing a handwritten letter or poem make people feel cowardly and hypocritical?

No, on the contrary, a handwritten letter will make people feel your solemnity and sincerity, which is extremely precious in today's era of overflowing electronic information.

You may not be extremely accomplished in calligraphy, but as long as your handwriting is neat and the paper is neat, you can still leave a good impression. Poems written with pen and paper often have their own aura, and those who like to write and read poetry will never go out of style.