If you can see the play it is too much!
I suspect that the link that is bad is the cheap one. They are $31 a piece. The other three links are $48, $61, and $65 a piece. They are all relatively easy to replace. All four on both sides would list for 410 if my calculator is functioning.
As to whether 17 inch tires would cause the problem... obviously! The same part is used on the E320 with 16 inch wheels and 215 tires. Its actually the same part that is current number for the 124 body 300E which used 195 tires on 15 inch rims. If you build a car like a race car or drive one like a race car parts don't last as long. The loading changes immensely.
Bosch Master, ASE Master, L1
33 years MB technician