Only 675LB & no stairs??  Hell, that is almost one man size!  The appliance cart will handle that easily.  I move boilers heavier than that on a $19 two wheeler from Odd Lots using a $2.00 ratchet strap.  (Local Chinese Junk emporium.)  It is amazing what you can move on cardboard or pipes.  I moved my 5000++ LB safe into my garage on pipes.  Just put one in the gun store that goes about 4000 LB and we never broke a sweat - either one of us.  (We cheated; used some equipment.)

When moving something a bit heavy, THINK about what you are doing.  Move VERY slowly.  PLAN every move.  ONE person is in charge.  STOP means just that!!!!!  NEVER put a body part under a heavy object.  NEVER.  Do NOT allow fingers etc to get in "pinch points".  ASSUME the object weighs much more than it does.  MEASURE TWICE, move once.  NO BEER until AFTER the move!!  If it tests your strength, you ain't doin it right!  Use your BRAIN, not your BACK.

I move boilers for a living.  They start at 450 LB and go WAY up from there.  Most are installed in basements.  Old ones come OUT of those basements.  So far (21 years) I have done $50 damage to one house and still zero to my body and my helper.  Yea, just two of us do it.  I sometimes move grocery store equipment like display cases.  Run about a ton or so.  You can't believe how easy it is with a "mule" dolly and some bread carts.  When available, case jacks are the shit.  They work on safes too.  Piano dollies are similar.  Would work well on your little safe; they rent for a small sum.  MAY be easier for an amateur than appliance cart.

ABOVE ALL REMEMBER THAT - SAFETY IS #1!!!!!!!!!!