No, increasing the injury rating will not bring them back quicker. The injury rating primarily sets how often a player is injured, the toughness rating dictates the length of the injury (based on the type of injury). Once the injury occurs, the length is set, except for this year if you get a Big Decision for the injury.
When they come back from injury, your trainer level dictates the level of regression including a possible reduction in injury rating. So you will want to progress the player to make up some of the regression and avoid being injured again in the future.
This is also why being an owner is better as you can hire the best trainer, as a coach, I believe the default trainer stays in place therefore you cant control the level of injury regression.
E.g. Vikings have a default world class trainer, therefore with Teddy Bridgewater's new injury, when he returns in CFM year 2, he will regress due to failing at season goals and regress (slightly) due to recovering from the injury. Had the default trainer been worse, his regression would be worse. I will run the sim when the new roster drops, but his INJ rating of 92 should also go down. If the season goal regression is severe I don't want to double down on his regression so will probably switch him to FB before starting a new CFM to get the blanket "rushing yards" goal whilst on IR.