Design of an electronic device to disable remote access ability of contactless payment cards without visible alterations and damage to the card
Electronically: Produce a series tuned circuit tuned resonant at the frequency of the card system - this acts as a "suck out trap. This could be conductive ink printed on paper or similar and stuck on the card surface. The trap need not occupy the whole card area but mechanically overlapping part of the loop conductors may help.

I have not investigated the technology. I assume that this is NFC rather than RF technology - but such loops are almost always still resonant and the same key principles apply. 

A similar device tapped to a TV antenna ribbon cable feeder of yore could remove a selected station from the spectrum. 

Keeping the card in a shielded sleeve when not in use would be easy enough. This IS an electronic solution. 



Related:

Changing banks to one which is receptive (and which will provide cards which are not) is liable to be most effective overall. I'm not exceptionally paranoid (just more than some) but I would not happily have one of these cards if I could not enable and disable it readily at will. The opportunities for abuse are vast. 

Adding a 'shield' on the outer surface of a card over the antenna area would be liable to 
work. A piece of soft iron tape may work. A strip of wide magnetic recording tape (off old half inch tape or ...) may.  A mumetal strip may also work for the opposite reason. As all these could effectively present as or under a thin sticker thay may alter the cards cosmesis but not its internal integrity. They MAY make it too thick to work in a magnetic stripe reader depending on how implemented, but this will often not be an issue.

A part sleeve would still allow the card to be used in most ways while screening the antenna. This has the advantage of being removable as required. 

A method similar to the one you cite but far less defacing would be to drill one only small diameter hole to break one only track anywhere in the loop. This is not certain to work but probably would. Drilling this from one side only and not penetrating the other surface improves cosmetic effect. The obvious location in in the magnetic stripe, with the hole then being black filled in some manner. The end result could be near invisible and all the cards functions  except one would still operate correctly.

The image below has an ~= 1mm dia hole through a track in the antenna loop under the mag stripe at upper right.



Another option may be to impact the card with a sharpish edge of appropriate width at relevant locations one or more times in order to fracture the internal loop. This may or may not be able to be done without visibly severe damage to the card. 

A commenter here suggested a slit in from the edge. They suggested it as the mag stripe edge but it could be also from either end - with the left end as seen in XRay possibly being closest. The slit can be essentially zero width - eg a shearing action which rips the card in to the required depth from the edge but has no actual width would sever the tracks and then allow the card to re-align. This MAY be very low visibility and "healing" with superglue or another adhesive may work.