The SSR only offers a single normally open contact. So, it will only serve to replace one "make" switch and would require more than one if there were more NO switches on the relay stack. It could not replace make/brake (SPDT) switch. I certainly wouldn't bother.

Usually, the sound can be dampened with resurfacing the coil slug, armature, or just replacing the relay coil.

Also, for "always on" relays, Clay on Pinrepair.com discusses converting these relays into DC by changing the coil, and adding a diode and cap just like Gottlieb did at the very end of EMs.

As most Series relays are paired in the ohm rating with other relay coils, it would be difficult to replace it with anything but the original coil.

The second relay you link to would probably work. This would be the better choice over the SSR. But it would be considered a really bad hack. However, it is YOUR game and you can do what you want with it.

How about this solution from Clay-
http://www.pinrepair.com/em/index3.htm#continuous

I think the best solution is the original equipment. The relay contacts are usually very easy to access when servicing is required. A lot can be said to the basic design of any device lasting more than a century, and the nominal service life of +5 decades. I personally prefer the time tested and proven durability of an open relay versus a sealed or SS unit.

Don't expect the Series relay to ever be quiet. They buzz on all games I have played. I feel it is kind of the charm in playing one of the 60's Gottliebs.