Jet engines are fairly expensive, accounting for a large part of the cost of an airliner. So there will be some incentive to repair the engine if possible.
Whether an engine is repairable is dependent on the failure. If it happens at startup and mostly causes a bunch of smoke, like in the linked video, then the damage may be fairly limited and will probably be repaired. If damage is suspected, a borescope inspection is generally done to check the inner parts of the engine for damage as well. If a failure happens at full thrust, the engine is probably finished. Aircraft and engines are designed for this failure, but a full blade failure will cause extensive damage. Parts that get loose and enter the engine core will cause further damage through the engine.
Also, serious failures are generally investigated, especially if they are considered uncontained. This will mean at least extensive downtime for the engine, and may or may not end up being repaired.
There are many malfunctions that can occur at startup. The engine has been shut down for some period of time, during which:
- Parts cool down and contract
- Debris can enter the engine
- Things can leak or get damaged
Whatever has changed will be discovered when the engine is started. Also, the problem in the video seemed to only appear after taxiing for some distance, so the engine could have ingested FOD at some point. It's not clear whether the 10 minute hold is due to ATC or the pilots noticing abnormal engine parameters.