You may be correct in your assessment. I cannot say conclusively one way or another, only offer my opinion. Also, yeast is very hardy and resilient, and given enough time can make a full comeback - it's just a matter of if that will happen soon enough.
You brewed an all grain batch 3 days ago and pitched 1-month old yeastcake. In my experience, all grain wort has more particulate matter than extract (grain material, proteins, break material, etc). This material remains mostly unaffected prior to fermentation, and begins to settle given enough time and stagnicity (is that even a word
). During this time, you can actually see various-sized small particles floating around in the beer. During fermentation, this material is "pulverized" into much smaller particles (some is consumed by yeast) and ends up being extremely small, most of which settles into the final trub layer. After fermentation, you won't generally see the kind of sedimentation you're seeing in your carboy where the particles are stuck to the carboy basically from the bottom to the top of the carboy. That stuff is knocked off during fermentation and settles to the bottom; perhaps it's less sticky?
Assuming fermentation began almost immediately (within 1-3 hours), you could expect your beer to be mostly fermented by day 3, however I would not expect clarity of any of the beer (like your upper 2.5 gallons). Additionally, if fermentation began that quickly you certainly would have noticed the churning effect inside the carboy - it would looks like someone is using an electric mixer inside the carboy. Also, I would not expect flocculation like you're seeing for 7+ days (minimum) without manually cold crashing your fermenter.
_I_ would not wait longer than 72 hours before addressing a possible non-fermentation, and I certainly wouldn't wait 2 weeks given the facts you've provided. My belief is that the yeastcake you pitched had very little viability left in it (<40%; more likely <20%) and it's lag phase is currently taking place until yeast populations are reached or the oxygen is depleted. The more time that passes the more chance for bad bugs to do their bad work. Most people will not wait longer than 1-2 weeks (MAX) before reusing a stored yeastcake. If the time has been longer then most folks will create a starter 1-2 days in advance.
Lastly, given that your original recipe calls for Safale US05, I would suggest picking up a sachet and putting it in your fridge. It's probably <$4 and will last for 1-2 years in your fridge in case you should ever need it. If this beer ends up being fine or finally starting to ferment then simply use this sachet for your next batch, otherwise an additional $4 investment might just save a $20+ investment.