Options for installing sub-panel
Conduit is still your friend

Even though running rigid EMT back to the main panel from the basement junction box is basically impractical with all those other cables in the way, it is still possible to stay in conduit all the way back to the panel.  How? Flexible conduit, that's how!

In particular, instead of running a cable (or cables) back to the panel, we replace all the existing cabling between the panel and the basement JB with 3/4" FMC, using a 90&deg; fitting to come into the bottom of the basement JB as the EMT going up to the third floor leaves very little room to do anything else.  We can then pull wires as-needed through that FMC: 3 6AWG THHN feeder wires, 4 14AWG THHNs for the existing 15A branch circuits, and a 10AWG or 8AWG bare ground will all fit through a 3/4" FMC, although it will be a tight pull, or you could pull wires for the MWBC route through it if you preferred, using a 12AWG bare ground from the panel to the basement JB.  (The reason for the ground wire here is that FMC isn't allowed to be a grounding conductor by itself when more than 6' of it is used in a run.)

And yes, 6AWG is (more than) fine for a 60A subpanel

Anyone who is telling you that 6AWG is insufficient for 60A is full of hooey, as a 6AWG THHN copper wire landed on 75&deg;C connectors and terminations (all modern distribution-equipment terminations are slash rated 60&deg;C/75&deg;C, which means that 110.14(C)(1)'s 60&deg;C termination limit does not apply to them), and otherwise allowed to run at 75&deg;C (as a wire in conduit is), is good for 65A.  (In fact, thanks to the 240.4(B) round-up rule, you can use it for a 70A subpanel, even, although there's no point to that in your case.)

The only caveat for this is that the wire connectors used in the junction boxes must also be rated for 75&deg;C usage.  This is true of most mechanical set-screw (Polaris&trade; or equivalent) connectors, as well as of decent quality wire-nuts, but may not be true in all cases, so do check the specs on what you're using.

GO BIG OR GO HOME

When it comes to subpanel shopping, you will want to go big here.  A 24-space or 30-space, 125A, main lug subpanel is not at all out of place here; if you wished to stick with GE, a TLM2412CCU (if you don't mind grey) or TLM2412CCUW (if "default electrical panel grey" would be a bothersome edition to the decor; the W suffix stands for a unit that's factory painted white) with a TGK32 ground bar field-fitted would be a good choice here.

As to getting wires from the third floor junction box into the subpanel, I would use a LB, accessed from the opposite side of the wall, and a 3/4" EMT run inside the stud bay to come down into the top of the panel.  This will require a bit of patchwork, but gives you a 100% conduit installation while letting you flush mount the panel.  Of course, if you'd rather have the panel surface mounted, that's an option as well, clear space permitting, and would let you simply run a length of 3/4" EMT down from the third floor junction box to the top of the panel.