Confirming the set-up and wiring of 60AMP subpanel
Your main panel

I gather your breaker in the upper left corner of the main panel, yes? Your breaker is back-fed, and is 60A.  Since your main is the same rating as your subpanel feeder, you don't actually need a feed breaker in the main panel.  You can just grab the subfeed lugs at the top of the panel.  That will free up 2 spaces. 

You will need the 2 spaces if you upsize the main panel's main breaker to 125A, but then, you'll also need a 60A feed breaker at that point.  At that point you will hit stab limits on the stabs that the main breaker is clamped to; i.e. You won't be able to have any breakers across from the main breaker because the stabs can only carry 125A, and the main will absorb it all. However, those forbidden spaces can be used for a generator interlock, if you had any thoughts to that. 

Three of your breakers look like Siemens, Murray or ITE.  Those are alien breakers in this panel, and they must go, presuming this is a GE panel.  They will not fit the bus properly.  The only non-GE breaker I am aware of that can fit that panel is Eaton CL. If you need shunt-trip or remote on/off capability, Eaton CL has that. 

Grounds

They make smaller ground bars that may fit in the ample wire bending space at the top and bottom.  All neutrals should be brought to their breaker for AFCI/GFCI reasons.  All wires should be cut long enough to reach any location in the panel. 

If you can run the conduit entirely in EMT, you don't need a ground wire.  However if you have one, think future and go #8.  You might upsize the wire in the future, and why buy the ground wire twice?  

Your subpanel does not need a main breaker if it's in the same building.  If it is in an outbuilding, you need a disconnect switch, and the breaker will suffice but its size doen't matter.  So there is no earthly reason to change out that subpanel main breaker. 

Getting the most out of conduit

Conduit fill: you can get more than 60A out of #6 in conduit.  


You can run THHN at the 75C column (65A rounding up to 70A breaker) if terminations at both ends are rated for 75C, which breakers are.   
However you are capped off at 80% of its 90C rating (75A)  giving 60A, because you have other circuits in the pipe.  


If you can find a way to get those other two circuits out of that pipe, you can use a 70A breaker on the feed.  

I would also consider larger than 3/4" sized conduit, to empower a switch to aluminum wire and future upsize. At 60A+ copper is a waste of money. Aluminum's bad reputation is inapplicable to these large sizes; and an AA-1350 feeder would be perfectly fine.  That's outlawed today; now you get AA-8000.