If the bone loss is mild then yes, bone grafting can be tried without extracting the tooth and sometimes splinting can help the tooth move less, which helps in healing
But But if the bone loss is too much then bone grafting alone is unlikely to save the tooth and extraction + replacement (bridge or implant)
In your case, It's grade 1 mobility so it can be saved without extraction But also it needs further evaluation like Radiograph and periodontal pocket depth (Gums)
Next Steps
you should go for
1. Deep cleaning (scaling/root planing)
2. Possibly guided tissue/bone regeneration surgery
3. Bone graft placement around the roots & then Splinting the mobile teeth for stability during healing
Health Tips
This isn’t guaranteed to work in every situation, especially if:
1. Bone loss is extensive
2. There are deep periodontal pockets
3. Gum disease isn’t well controlled