They do. Ultrasound in orthopedics is a relatively newer field, and there aren't quite as many sonography techs and radiologists experienced in reading these studies, which is likely why you don't see it offered more widely.

Edit: I should mention that ultrasound is basically unusable for evaluating bones. Sound waves can't penetrate bone, and so you end up just seeing a huge black void. That's a huge orthopedics use case that ultrasound just can't benefit. However, ultrasound is fantastic for evaluating muscles, ligaments, tendons, and other superficial soft tissues.

We order ultrasounds all the time for shoulders (for like soft tissue issues; for trauma, you'd start with an xray). For other joints, such as the knee, MRIs are a better choice (unless htere has been substantial trauma, in which case xray initially or further), though more expensive, unless you're excluding a Baker's cyst, in which case an ultrasound is fine.

Since MRIs are more expensive, private doctor's might order them instead of an ultrasounds.

(I'm a doctor)

It's a manual, non-standardized process without a standardized output. Image quality depends both on user skills (how deeply they press the sensor on the skin) and the machine they have. Unlike CT/MRI the examination results cannot be easily shared and compared between patients for studies.