First.  Welcome.  This is a great forum that has helped me bring 3 Savages back to life.  You're in good hands.  There are many members with more experience than myself, but here are a few basic things to consider.

Seafoam in small amounts is fine.  It doesn't work miracles though and it's better for maintenance and light duty.  Let's not jump ahead yet to the possibility of a clogged pilot jet/dirty carb.  Hold that thought for now.

When checking the oil, the bike need to be straight up and down (perfectly vertical).  The bike can't be leaning on the side stand.  Get a buddy or somebody to sit on it straight up and down while you look.  Or, learn to hug the bike by the seat area to get it vertical from the right side while you try to see the sight glass window.  Or, if you're young and real flexible, you can sit on the bike and contort yourself to twist and look down and see the sight glass.

You have enough oil in there for it to start.  A new filter won't make it start.  Get the oil to the right fill level, but this isn't the main problem.

When you put it on "Prime", leave it there.  You can run the bike all day long on Prime.

Some clarification - The bike actually turns over (it's trying).  You can hear the starter turning when you hit the start button.  Right?  You can hear it trying to start?  Right?  

IF it doesn't make a peep (won't turn over) when you hit the starter button, it's at least partly an electrical problem.

It does seem fuel related if its "trying", but it could be spark related.  You basicially aren't getting combustion.

Try leaving the petcock valve on Prime. Don't twist the throttle while cranking it. 

If you have to try multiple times, only hit the starter button for like 5-8 seconds max (2-3 seconds is safer).  Stop.  Let the starter rest and cool off.  Wait 5-10 minutes and try again.  Be patient - you don't want to burn up the starter motor holding the starter button down for like 30 seconds and then impatiently trying again 2 seconds later.

You can test the petcock valve and make sure it's not clogged by putting it on Prime and pulling the fuel line off at the carb.  If gas come out the fuel line and spills all over your hands/rag, you know its not completely clogged and the carburetor is getting fuel to it.  If you don't get fuel coming out, you have a petcock valve problem.  There is a filter/screen on the petcock valve inside the gas tank (it can get clogged with crap)  Or the guts inside the petcock valve can fail (stuck shut).  Get a Raptor petcock valve if you find your petcock valve isn't woring correctly.  

More about the petcock valve here...  -->  https://suzukisavage.com/cgi-bin/YaBB.pl?num=1251932429/1

I see my confusion with the turn over, I guess better phrasing was crank-but-no-start for my specifics, I appreciate that clarification.

I should have also noted the battery is new (literally bought yesterday) - I installed a battery tender harness so I could plug in a battery tender and I made sure the battery terminals were in contact with the bikes ring terminals before the tender harness. I think it would be reasonable to assume a low battery is not the cause, but I will test anyway for general practice.

Luckily I got a spark plug tester so I can check that easy as well.

Thanks for all the info regarding the petcock!

To build on JOG's message...

Starting Fluid (ether) is fine.  Instead of removing the seat, you can remove the side tin and air filter and direct the starting fluid into the snorkel tube using the straw included with the starting fluid.  You will want to open the butterfly on the carb to get the starting fluid past it and into, or as close as possible, to the intake manifold as possible.

It's actually easier to test the petcock valve and make sure it's delivering fuel to the carb.  If the carb is getting fuel and it still won't start, I would guess a stuck float valve or dirty pilot jet in the carb.  You could then try the starting fluid method and see if you can make it go bang.  If it'll go bang with the starting fluid, you can pretty much determine its a dirty carb (float valve stuck closed or clogged pilot jet).  If it's a stuck float valve, you could try tapping the float bowl of the carb with the handle of a screwdriver (not a hammer).

Others have given good advice. Better than I can give. You need to cover your bases. Start checking things.  Spark and Fuel are super easy to check. Spark plugs and plug wire are cheap too. Hopefully your issue is a simple plug and plug wire swap or something like that. To me it sounds like fuel or spark issue.

Also, get some JIS screw drivers. $20 set from Vessel on amazon. I believe it's full tang. It has been treating me great working on these Japanese bikes.