IMHO you did the best thing possible. although you exposed the sensitive nail bed by cutting away the nail, you allowed excellent access for cleaning - which is the objective.

a caution with using a syringe - you run the risk of forcing dirt & bacteria further under / into a wound - unless you use a lot of water and can control the direction and flow of water. a bag is great.

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“Everyone should have a horse. It is a great way to store meat without refrigeration. Just don’t ever get on one.”
- ponder's dad