Can you use ECDSA on pairing-friendly curves?
You can. Low-embedding degree may be bad due to the MOV attack, but pairing-friendly curves are particularly chosen so that the embedding degree is low but still enough to not decrease security. So any elliptic curve algorithm should be safe on the curve, not only pairing-based ones.
Some observations:

ECDSA if often used with NIST curves with cofactor 1. However, some pairing-friendly curves have larger cofactors, so you need to take that into account. For example, the BLS12-381 curve has a 381-bit prime but the main subgroup has 255 bits, therefore is has around $255/2 \approx 128$ bits of security (and not $381/2$)
It's still a bit riskier than using high-embedding degree curves. If there are advances in attacks against discrete logs in finite fields then the curve security can decrease. In fact, this has already happened once: for example, the BN256 curve security was reduced from 128 bits to 110 bits.