Usually when the input shaft fails, they refer to the teeth that drive the counter gear shearing off. When this happens, you lose every gear except 4th gear. Now grinding into 4th could mean the dog teeth on the input shaft are worn. This would mean you need a new one since the teeth are part of the shaft
Also, if this shop knew anything about Mustang trans, they would know to swap out the 8 tooth speedo drive gear with a 7 tooth speedo drive gear and run a red 21-tooth speed gear. This is what you do to avoid running the white 23-tooth gear...which will shear a lot because the teeth are so thin. Anyone who has a 90+ T-5 apart should replace their 8 tooth with a 7 tooth automatically.
Popping out of first means bad dog teeth on the gears. Grinding is usually worn synchro blocker rings.
New 1st gear = $85
new 1-2 slider = $85
Typical T-5 rebuild kit = $200
New Input Shaft = $150
That's about $550 in parts. Toss in another $100 for markup.
That's about $750 for labor which includes pulling and installing the trans.
SOunds expensive, but that's about right.