The way that the accompaniment to a vocal exercise is played matters a lot. The biggest problem I hear with young or insensitive teachers is rushing. The second biggest problem is probably excessive volume. Fixing these two problems does not depend on being an expert pianist.
Sometimes playing the introduction loudly helps a singer to sing out more, if needed. However, while the singer is singing, the accompaniment should stay out of the way, usually.
Some things to try:
If the exercise is to be repeated at different pitch levels, try subtracting notes after each iteration until you are just sounding a chord for the singer, and let the singer do all the moving notes while you hold the chord.
Then try subtracting notes from the chord, if the singer can sing the whole exercise with just a starting pitch.
Try accompanying in a different octave from the singer.
Try playing both hands with the singer (unison octaves).
If you are playing 1358531 patterns in the right hand, experiment with playing just the tonic in the left hand, and releasing it halfway through each iteration, or go down to the fifth (in contrary motion from the singer) when the singer reaches the top of the arpeggio, returning to tonic with the singer.
Play with NO pedal most of the time.
Play softer than you think you should. Then softer than that.
If you are asking for a dynamic change within an exercise, get creative about supporting that – adding a tremolo that changes in volume with the singer, stacking up harmonies with the pedal or thinning them out, playing thicker harmonies for forte, and lean ones for piano.
Talk with the singer about the liquid flowing legato and portamento possible with the voice and how that contrasts with the terraced pitches and decaying dynamics of the piano. Play staccato against their legato and vice versa.
Consider mixing up the rhythm of the accompaniments. It doesn’t have to match the singer. Try oom-pah for 4/4 and oom-pah-pah for triple time.
Run some exercises diatonically rather than chromatically, and harmonize each step.
In scalar exercises, play every other note in the right hand, or just the 1st, 5th, 9th, etc. notes. Do this especially with fast scales so that you and the singer can hear whether true scale steps are being sung. In 12345678987654321 play the I chord for the ascent and the V chord for the descent.
Play the singer’s pattern in the left hand and harmonize above with the right.
Play in such a manner that you can always hear the singer clearly. Play patterns you can handle in any key, and if you have any limitations playing simple accompaniments, work to overcome them.
Why bother with any of this? Because banging out arpeggios and five-note scales chromatically all the time is not musical or interesting, and does not support the singer’s ability to hold their part independently or to hear themselves. Exercises can have many musical elements, if we let them. The last thing we want to do is go into an immediate trance through dull routine.