Can Static Destroy Electronics?

Electrostatic discharge (ESD) can absolutely damage electronics - this video (you can download a pdf of slides) by Texas Instruments (experts and reputable source) details the issue -  TI Precision Labs - Op Amps: Electrostatic Discharge (ESD) | TI.com Video.  BUT, a lot has to do with the individual design - how good were the components that were used, and how many times has the circuit been subjected to the ESD.  Subject op-amps to repeated ESD events and eventually they can fail.  

To the OP, using an anti-static brush incorrectly is an bad as not using it all.  Assuming you are using the common Audioquest carbon brush, for that to really work you need to ground yourself (touch any metal that has an electrical ground) to discharge any static you and/or the record have to ground.  You did not state - but is the tonearm grounded to the phono-amp?  If you have a multi-meter, you can easily measure resistance between all components (metal casing) and the wall outlet ground (power source) to verify all are grounded.  

ESD mats generally have two wire connections, one from the mat to earth ground such as the screw that holds a wall outlet plate, and the other is a conductive wrist strap that grounds you to the matt/ to the wall plate.  If you put a matt down on the ground/carpet  that is grounded (to the wall plate) and you are wearing shoes that are not conductive - its not going to work.  Either you are barefoot, or your shoes have to be conductive to provide a ground path for you.  You could instead use a touch pad such as the following -  Amazon.com: Anti-Static Touch Pad Touch Pad & Grounding Cord: Home & Kitchen.

FWIW, I use an ESD pad for my table platter mat - but its not for grounding me (but if I touch my metal platter it will ground me), and if you are interested the details are here with discussion and links for reports/test on record static http://www.vpiforum.com/viewtopic.php?f=7&t=17186#p65882

There is an article on the grounding of the Technics SL-1200 and you can find it here -  Internal Grounding » Technics 1200 Parts, Technics 1210 Parts, Technics Repair Turntable Repair, Mods Modifications, Pro Audio Parts, Accessories, and Repairs, Toronto, Ontario, Canada (technics1200s.com).  It appears that the newer units are no longer internally grounded.  Its not a safety issue, many electrical equipment such as plastic housing power drills are only 2-wire; from a ground safety - they are intrinsically safe and the reputable ones are UL approved.  But even if the motor was grounded, unless the metal tonearm is somehow connected, grounding the motor is not going to help.