There are several things that can cause similar symptoms, they are relatively easy to check and cost nothing to test.
Is the transmission full to the correct level ? With the correct fluid type ?
Is the Throttle position Sensor and the Neutral Safety Switch functioning correctly ?
Is the kickdown linkage properly adjusted ?
Have you tested the shift solenoids ?
Stupid answer is not necessarily stupid, in fact that answer may be more logical than yours.
For starters the fluid level is important but highly unlikely to cause OP's concern, fluid type is irrelevant here. The TP sensor has absolutely nothing to do with the OP's concern, or would not affect shifts at all from 1-3. The neutral safety switch will not have any affect on shifting for this trans. Also there is no "kickdown" linkage, it is T/V (throttle valve) linkage, this linkage is also important to be there, functioning and adjusted properly but still would not cause the OP's concern. And last, there is no shift solenoids inside this trans that would cause any problems in the gears mentioned.
OP, from what you describe it sounds like the shop diagnosis is probably correct and it's missing 2nd gear. The reason for high RPM's for upshifts is probably because you are needing to build up enough MPH to shift into 3rd from 1st, missing 2nd totally. 2nd gear is hydraulically controlled, and its main component is from the intermediate band applying.
Most likely you have a problem in the intermediate band (aka, front, 2nd or kickdown band) being broken, worn or burned, also possible to have a servo, band strut, valve body or hydraulic problem too, there is not telling. Unless you know what to look there is not much else to consider without dropping the pan and possibly the valve body for further inspection. Its time to consider or assume a rebuild, new/reman and the bare minimum being removal, partial tear down and fix.