Apparently all roughed out bows of this type were fired using very heavy arrows to break them in and uncover any problems then the bow was sent to a skilled craftsman who shortened the stave and finished the bow out to best suit the owner.
You're mixing up Roger Ascham's advice for a personal bow written in Toxophilus with a livery issued military bow. There is no evidence whatsoever to suggest that any of the MR bows were unfinished and waiting to be retillered to suit a particular archer. Ascham recommends that when buying your own bow for personal use, to shoot it in with heavy arrows then having a bowyer pike it to bring it into it's final, consistent and reliable form.
I went by Saxton Pope's analysis, he examining the only two bows that were recovered up to that point. The first two bows, more properly bow staves, were raised by hard hat divers just after the ship was located in 1836. The chests of completed bows were found till more than a century later.
Those staves were 6 feet 4 3/4 inches each, which proved to be too long to be efficient resulting in a low draw weight and sluggish performance. If used as they were they would have been practically useless.
Since by law every man between 16 and 60 , with few exceptions, were required to obtain and keep a long bow and at minimum 12 arrows ready to go its not likely that professional archers would leave the bows they had paid good money for behind to use whatever happened to be handy.
The MR bows were complete bows,
Those raised in the 20th century were, as near as they can tell. Not all have been fully examined even now.
There's no point sending a warship out into an immediate naval battle full of bows that aren't ready for use,
No one said they did, the two unfinished staves were the exceptions. It would be unusual for such a vessel to have no armorers aboard ready to repair or replace damaged arms.
All the bows not found in the wreckage itself rotted away centuries ago, so there's no way of knowing if any of the bows in the chests had been issued to bowmen aboard the ship or if they were cargo.
The Mary Rose went down in the heat of battle. The bowmen aboard would have had plenty of time before battle was joined to claim their bows and be standing at the ready.
As an aside, the comment that not all bowmen could shoot 120lb is probably slightly unlikely - most healthy men today can shoot 100lb-120lb with a couple of years training,
I rather doubt that. Perhaps most athletes could. Most men of that time period would not have had arms long enough to bring a 120 lb bow back to full draw without losing the leverage necessary to do so.
Last I heard the recovered completed MR bows that have been closely examined would have varied in weight from 80-120 lb and apparently one replica was constructed that has a 160 lb draw.
If every man could handle the 120+ lb bows then all the bows found aboard would have been of 120 lbs and up.
Long bows stored in arms chests in the hold of a ship aren't at the ready to be handed out. They didn't just toss a pile of bows on the deck and say grab one and go to it. Closing speeds of ships in those days meant it could take hours or even days before ships came into range. Plenty of time to form up and choose your weaponry. Not every soldier much less every crewman aboard could effectively use a bow at all despite having trained since youth. Some were armed with pole arms, some were gunners, some were sail makers etc.
Another thing to consider is that not only would an archer have to draw a bow once , during a battle he would be expected to fire volley after volley. What a modern archer may do under ideal conditions is no gauge of what a bowman in battle must be able to do several times per minute perhaps dozens of times per hour, hour after hour.
Nope, I figure most bowmen of the day could handle a LB with 80 lb pull all day long, but only the biggest and strongest could effectively use bows of 100-120 pounds and a rare few could handle the very rare bows of over 120 to 160 pounds.
Bows of 80-100 lb were suited to the common levies while anything heavier were for use of the fittest and most experienced professionals.