I would say yes, and yes. You've explained the problem pretty clearly, and explained its consequence. Choirs frequently find that they sing everything internally, consistently in-tune throughout a piece, but then at the end of the piece, they discover that they are no longer in tune with the reference pitches upon which they started the piece. The frame of reference has drifted. Inexperienced singers don't understand that this is in fact a natural occurance. In fact many choir directors don't understand it either, and expect their choirs to sing in 12-tone equal temperament as if there was a piano playing along all the time.
However, singers and string players can compensate for the drift in pitch and try to keep pitches in line with the starting pitches, and end the piece in the same pitch framework where they started. It is a struggle, but it is often possible.
Reference this previously asked question, about Arnold Schoenberg's observation about why choirs get "off pitch": Natural vs. Tempered Semi-Tones
As I said previously in my answer to that question:
As a singer with perfect pitch, when I am in this situation,
particularly in performance, it presents a dilemma: if I perceive the
other choral singers around me drifting away from the starting pitch,
tonic and key (taking into account the modulations that may be written
into the piece), do I fight against the other singers by keeping my
pitches up, relative to the starting pitch and tonic and key, or do I
"go with it" and follow the gradual shift in tonic that everyone else
is making? In most cases, it's better to "go with it" rather than
"fight it", because it produces a more internally consistent sound.
While I have perfect pitch, I must concede that if a choir were only
to sing the 12 pitches on the piano, that would not be a good choir,
because that is not really what choirs are supposed to do.