Do people need to be empathic in order to care or have concern? No. And, sometimes that empathy can even potentially lead to less care. But, what is most important for you? Would you prefer someone who is caring, but may not understand what you are feeling/thinking or do you prefer someone who understands you? Hopefully your partner (or future partner) both can understand AND care and respond appropriately to the emotions you express (and hopefully YOU are that future partner as well).
Citation: Winczewski, L. A., Bowen, J. D., & Collins, N. L. (2016). Is empathic accuracy enough to facilitate responsive behavior in dyadic interaction? Distinguishing ability from motivation. Psychological Science, 27(3), 394-404. https://doi.org/10.1177/0956797615624491