Help needed with mounting sink to wall
Question: Want to know if lag screws will be solid enough to hold the sink to the wall. I originally bought brackets (picture attached), but I am unsure that I need them.

Yes – lag screws would be able to support the weight. And the use of the bracket depends on the points I bring up when installing this sink as a utility sink.
Your utility sink was free standing and most likely had a splash panel on the back or the area behind it was protected from water. The drywall was either backer board or drywall cover by appropriate layer to prevent water damage.  
Using the lag screws with appropriate washer is doable, if the proper support is provided on the right side as you stated.  You must be careful when tighten the lag screws enough to hold the sink securely WITHOUT overtighten that the sink or coating will crack.  A heavy rubber gasket between two washers can be used, but again do not over tighten.
Use of the brackets:  After securing the two brackets flushed to the surface with appropriate long heavy screws (3” deck screws/) to the support, you will be able to just lift the sink and hook the sink to the two brackets.  The part on the left of your picture are L hooks.  Push the sink over the hooks, and the hooks will garb the sink. Using the brackets, you do not have to worry about over-tighten the screws.
Using either method of attachment, you can tile or use other design to create your splash panel above or around your sink.