Looking for preamp which is good at BOTH single-ended AND balanced output

I think the important thing is to buy the best possible preamp (with both)… the sound quality of the preamp is critical… in many ways the most important component in your system… the difference between the two in/outputs are likely to be small. Given your price restrictions, consider used. You could reach into the next echelon at the $5k level that way. Need to get a preamp which has a sound you really find appealing. For me that has always been Audio Research or Conrad Johnson. You will need to discover the brand that speaks to you. Listen to all you can.

My equipment is Audio Research. It supports both single ended and balanced. The designs are balance, ARC recommends using balanced. I have compared both… the difference is just plain minuscule. The differences between different component brands or models  far exceeds the difference between the outputs on the same ARC component.

While no absolute guarantee, the higher the quality preamp the better both are likely to be and also the less the difference is likely to be.