ranjit, singing the actual pitches changes the exercise quite a lot: you are connecting the notation to a sound, rather than only naming notes in rhythm. I would distinguish that from the concern about feeling obliged to verbalise every note while playing.
It also helps to separate fixed-do and movable-do. With movable-do, G–A–B in G major is do–re–mi; move the phrase to C major and the syllables stay the same. That is useful for hearing scale-degree relationships, while finding the correct physical piano keys remains a separate skill. Movable-do is an established sight-singing approach, including in Berklee's Solfege 1 course, rather than something restricted to a few children's tunes.
A practical check would be a short phrase sung once on syllables, then on a neutral vowel, then played from the score. If the melody stays clear when the labels disappear, the syllables are serving as a learning aid rather than an extra step you must always perform.