What skills must cyclists master to succeed at racing? In addition to power, stamina, endurance, what skills must you master to succeed in professional cycling?
If you are not familiar with road cycling it may seem like simply a feat of strength/endurance, but there are also a number of skills required.

Skill: Group Riding

Riding in a peloton at speed around corners is difficult.  Often you can be at 65km/hr with 100+ other cyclists around you with only inches to spare while entering a series of corners.  In this situation you often get contact and there is pushing.  You have to be strong on the bike so that you can push back or take a hit without crashing.  This is not easy.  Sometimes opposing riders with deliberately nudge your handle bars in a corner to push you out of position, you need good balance to absorb and push back to hold your position.  You also only have inches to do this before you hit someone else in the peloton and potentially start a crash.

You also need to anticipate how the group moves, if you react too slow you will quickly filter to the back to the group, which is the worst place to be (riders in the back often get yo-yo'd as the group surges and slows).  You will then have to make repeated efforts to get back in or to move up. You can save a lot of energy and effort if you ride smart within the group.  This means constantly making micro tactical decisions.  Over the course of 200+ km race, all these little decisions can mean the difference between whether or not you can put in that last big effort to break from the group and take the win (a la Vino).  

Such a constant effort of positional awareness takes a lot of continued focus (our next skill).

Note - I know you said avoid discussion of tactics, but you also said tactics in terms of how the stage is progressing.  Before you can get to that point there are quite a few essential group riding tactics you need to learn.

Skill: Focus

Road racing is about being able to endure pain.  The more pain you can endure the faster you can perform.  There have been points where I thought I was going to have a heart attack because I was pushing so hard. You have to be able focus through this or you will lose your position and potential opportunities.  A good practical example is closing gaps.  Often breaks will get off the front of the peloton and if you want to join you have to jump the gap.  This can be very hard to do when the peloton is at speed.  It requires commitment and focus to sustain the effort needed to get there, as this will typically be an anaerobic effort.  

Then once you close the gap you need to focus again on damage control.  You just did a huge anaerobic effort, you can't breath, and all you want to do is rest, but the group you just caught has collectively decided to jump again.  You need strong focus to work through this and stay in the draft of the break group you just caught.  If your focus slips for even a second a gap can form, meaning you will lose the draft and the gap will widen.  You will then either need to do another anaerobic effort to crawl back in (even more focus) or you will fall out of the lead group and get eaten up by the peloton. 

Losing focus for a split second at the wrong time in a 3-4 hr race can mean the end to your day. Lets say for example your concentration waned and you fell out of that lead group.  You now lost your draft and are in no-man's land.  The peloton is now fast approaching and actually surges as it catches you.  Before you know what happens you are going straight through the peloton and out the back before.  Now you are off the back of the peloton, with no hopes of crawling back in.  Your day is done, you have just pulled a hero to zero because your concentration waned for a few split seconds.

Other sports such as Baseball and golf also require great feats of intense concentration.  The exact type of concentration differs, but if you cannot focus you will not be a strong racer.

Skill: Knowing your body

This may seems silly to mention, but professional cyclists are incredibly in-tune with how their body functions.  Because races are so long you have to pre-emptively eat and drink.  Waiting unit you are hungry or thirsty is too late, by this point your metabolism will be slowing down and will take time to recover meaning a period of reduced performance. Eat and drink too much and you will feel bloated or sick.  It is a difficult job to know the exact amount to eat and drink and when.  This skill often takes a number of years to hon.

Skill: Being Anal Retentive

Road cyclists are often made fun of for being anal retentive, and it is a deserved title.  That said, it is also a very necessary skill.  Riding a bike is not a "natural" activity.  We didn't evolve to cycle, as such you have to train and practice to be good.  The better the training and knowledge of the finer details the faster you will go.  This can be everything from muscle recruitment (how you activate your leg muscles and core muscle through the pedal stroke), to your position on the bike, to having your equipment in optimal working form (I swear cleaning and keeping the derailleur pulleys in good working order can gain you 20 seconds in a 10km TT).  

A strong performance often happens when you get all the ducks lined up (including tactics).  To do this you have to have a little streak (or a lot) of anal retentiveness.