Kick starter "teeth" not engaging now

[Restoration Fan]

My son and I decided to test the compression on the 78 CB750K he is rebuilding for his senior project.

When we purchased the bike, the guy we bought it from told us that the kick starter would "miss" about every now & then.  He kicked the bike over about 5 times and it missed one of those times.

As we were testing the compression on cylinder #1, I kicked it and it fired...same on 2nd, 3rd and 4th kick.  It missed on the 5th kick but fired again on the 6th kick.  At that point, it started missing every single time and is now no longer turning the engine over at all.  

When you kick it, it goes straight down and never seems to make the starter pawl or starter flange engage the starter clutch.  The good news was that the compression was 110 on cylinder #1 and still climbing when the bike stopped turning over.  The bad news is that I can't continue the compression test on that or any other cylinder now.

BUT, the kick starter does return right back up, so I feel like the starter spring is still engaged because it's still working.

We started to remove the clutch cover to try and see if we could see what is wrong in there, but 3 of the screws (yes, it has Philips head screws on it) are stripping.  I have PB Blaster on it right now and will be trying to use a screw extractor on it in the morning to get those 3 screws off and get the cover off.

My question is this:

The Honda Service Repair Manual says for disassembly that you need to first "Disassemble the cylinder head, and cam shaft tensioner in accordance with section 3-3b on page 32~35."

Do you really have to do that to disassemble the cylinder head and cam shaft tensioner to replace a broken piece?  (Like if it's the starter clutch missing some teeth or kick starter spawl or kick starter flange?)

[TOOLS1]

First, you do not do a compression test with the engine running. However I doubt you have a compression issue, and would not worry about it. Second to repair the kick starter, you must split the case. You can do this without removing the cylinder, and head, by just turning the engine over, removing the side covers, and assorted parts. Then remove the lower half of the crankcase. Now to remove those screws, take a dremel tool with a cut-off wheel, and carefully cut slots in the heads of the screws. Then use an impact driver (the one you hit with a hammer) to remove the screws. I have never seen anyone successfully remove a small screw with an easy-out.
TOOLS

[Restoration Fan]

Sorry, I didn't mean to imply that we had the engine running.  We removed the spark plugs and had the compression gauge in cylinder #1.  Turned the ignition key to on position (doesn't matter...battery is dead) and twisted the throttle completely open while kicking the kick starter.  When I say it "turned over", I meant that the kick starter gear was catching and causing compression in the cylinder.  On the times that it 'missed', it just went straight to the pegs like it was missing some teeth there or something.

Great idea about the dremel tool and impact driver.  I've actually removed some screws before with the screw extractors but I like the dremel idea much better before trying that.

Thank you much for the advice!