To the OP
I would recommend finding a hiking companion interested in going to the same places (peaks) as you and going together. The process of learning off trail travel is enhanced when there is someone to discuss things with. If they know more than you they can teach you but best to be sure they are correct. there is a special relationship (bond) that will develop if you learn together. You can bounce navigation questions off each other as you go and debrief afterwards basking in the glory of good trips and commiserating or laughing off the tough ones. ... The later has lead to some fine quotes such as : "Well, we could have done that better." "Bushwhacking is like garlic, it's an acquired taste". "How do you want to go back to the car? Anyway but the way we came up. I'm good with that."
I think prudence is warranted in the beginning as you build expertise at least for the sake of preservation of life. This speaks to the benefits of doing your homework such as your posting on this site and more granularly to the benefit of preparing for each hike as others have mentioned. After a while when skills are improved you can be more lax.
Navigation habits can be like personalities. Some people will hold to a bearing or route regardless of the terrain. Others will do a hike with only a few uses of their hard tools. The extremes can be quite entertaining to watch or quite painful/unnerving.
And also keep in mind that many people have climbed these peaks without every learning to navigate simply by following someone else. Such as a club. It is an option. And it can be a safe one depending on who you follow
. I've noticed an increase in the number of, let's say, low skilled organized groups. So beware.
Yes shorter routes can be safer but don't be deceived as more popular peaks or routes can offer less resistance. I thing everyone will agree that 1 mile of trail is easier than 0.2 miles of hellacious bushwhacking . I'm not saying that a 1 mile trail is better than a nice 1 mile bushwhack.
One point to note as you hike a HH peak if you follow a herd path you need to think like the herd. Meaning you might get to a tough spot where you scramble up a steep pitch only to find the path is not at the top. Most people would get to the pitch and find a route around it. So if you are following a herd path and get off it, IMO it is best to back up 20 feet and get back on it vs following the bearing. You need to be sure the path is headed where you want but that is part of the homework and a general compass bearing confirmation
Navigation is a passionate subject for many and there are many opinions and some nice pieces of advice already posted. I could +1 on many of them or spend hours typing my own but I won't, for now.
Enjoy the journey