Older 13 HP Briggs and Stratton conversation from points to electronic ignition

Ok HELP My Roper branded RT 13 has an ignition problem I cannot seam to solve and I'm at my witts end. I've rebuilt this machine and have way too much in to quit but I'm about to... So I lost spark at one point last year and replaced the condenser and points, then it lost spark again and I replaced the coil, then It lost spark again so I went back to the old condeser, after testing it, I figured the china junk wasn't working any more, old one tested out good new one was shot already after just a few runs, all good working again, washed it and lost spark again. WTF so I changed it all out to a new electronic coil no points or condeser required, started up and ran for about 15 minuted then made a horenous noise and died, pulled the head everything ok, but it was backfirring out the muffler so bad I thought it sucked in the exhaust valve, nope it sheared the flyway key and the timming was way off, replaced the key put it all back toegter for the umteenth time and it ran for a bout 20 minutes, set it to idle loaded my trailer with leaves, ran into the house to get a drink of water and as I stepped back out side it konked out, no problem just a restart, trottle it up and go again, nope, backfired out the exhaust really loud and some popping in the carburator, I take it apart again figuring its the keyway but it's fine? Hummmm.... Any help or suggestions greatly appreciated. I would replace the coil with a better quality one but have no way of telling the differnce between all the new ones out there, this is the one I have as suggested by a YouTube video but might not be right? So it's a 13 HP Briggs and Stratton.Model 302437 horizontal shaft motor. Open to suggestions... anything.

Ok I don't see my reply here so I am typing it in again. It has spark not so sure how strong it is but it back fires out the muffler with a good bang and pops in the carburetor as well, I can not imagine the valves have changed gap since it stalled out, I did pull the head and the valves were working properly and then it ran well again for 25 minutes or so, then it started misfiring again, the only thing I can figure is that the new electronic coil I choose is wrong somehow and I don't know how to research or verify it's the right one, or that it's working properly, I have tested the old coils and they tested OK but provided no spark, hence I thought it was the condenser again, points looked good, so I replaced the coil with the electronic coil to eliminate the points and condenser out of the equation and it worked like a charm until I sheared the flywheel keyway and then it worked agin until it stalled out while idling. I'd gladly replace the coils with a verified better quality one that fits the motor but I have yet to learn which one that might be. Yes I re-checked the keyway, it's good. I looked into the parts diagram but don't know how to find a cross reference number for a new electronic coil to replace this one. Thanks in advance for your help.
 

Tight valves will get worse as the engine heats up.
 
Backfiring thru the exhaust and/or the intake indicates valves not closing fully 
or timing off.
 
If you won't check the valve clearances, which isn't hard to do, do a leak down test.
If there's leaky valves, etc, it'll show.  
 
Re the coil, there should be some markings on the coil that states "UP" when mounting.
The kill wire tab is usually on the bottom, closest to the block.
 
.

Thesmallenginehobby guy I can get those pictures for you but not so sure what they will tell you or why the throttle has anything to do with timing and spark issues. "If you think it's electrical it's gas -- if you think it's gas it's and electrical issue" I get it, give me a few days to get the pics. I appreciate the help here guys -- I really don't know what to do, never had this problem before usually I can fix anything but this has me stumped.
 

Just curious but how are you checking for spark? Tried another spark plug? Spark plug fouling out?

Ok The weather and my time to work on it lately have not been in sync.
 
I do have a new spark plug, I changed it out several times during this just to rule it out.
 
I have a picture of the throttle set up, sort of, it is buried and hard to photograph but I will try to upload that later tonight possibly. It's a typical 1973 Horizontal shaft motor with an updraft carburetor throttle connection is mid linkage of governor rod, and a separate manual choke  
 
I have tried to run it without the kill switch wire on there, (especially when I initially look for spark during coil replacement) I have rung out all the wires and the switch, if those are malfunctioning then I would not expect any spark, yet I am getting backfiring through the muffler and carb like the timing is way off again.
 
I appreciate all the help and attention here, I sure need it, and will update everyone as soon as I can get back to it.

Yes exactly the one
Not sure how upload the pic but here's a try

Scott ;
I haven't had a chance to really work on it this weekend, or lately, I hope to get back to it tomorrow or Tuesday 
But if I were to guess "Yes" it gets spark. because it got spark when I installed and tested it before final assembly and then it ran for those 20 minutes or so, and then it must be sparking to cause the loud backfires in the muffler and the carb, but I will be checking for that as soon as I can put it back together, the flywheel nut is off from checking the keyway, which is still good, but had in the past operational running, sheared off, I am reluctant to assemble it back together at this point because I have done this 1000 times with limited sucsessful results, I was really hoping someone would see the errors of my ways and point out perhaps a good quality part that would instantly solve all my problems, yeah right huh? LOL But gauging by the line of questioning I will be re-assembling it all and running these tests. The upside is I am so good at teardown and reassembly now I could probably do it in my sleep in 2 seconds flat like a nascar pit crew with my eyes closed.Thanks again I will keep everyone posted.