Translation:Let's play soccer this weekend.
I think that's what OP was trying to point out? That both "on weekends" and "this weekend" are/should be acceptable translations.
However, I do think the use of the particle に with the invitational form implictly specifies "this weekend". Obviously it depends on context, but to me "on weekends" sounds more like 「週末は...」
遊ぶ（あそぶ）is "to play" by itself. As in; playing around, having fun, enjoying yourself. It's also used to mean "going out" (e.g. for drinking/karaoke).
When you 'play' a sport or instrument, Japanese uses a different verb. For most sports and activities (that I can currently think of) that's simply する "to do".
Yes, you can drop the に and replace it with a pause in speech or a 、 in writing, but be aware this makes it slightly more informal and may slightly alter the emphasis/implied meaning depending on the context.
As I've mentioned before in previous comments, 「週末は...」 instead of 「週末に...」 more strongly implies "on weekends" over "this weekend", and a pause allows the listener to infer a は or a に.
I had always thought that に was required in the same way that English requires particles in regards to time. At least, that's what my class taught me and I'm pretty positive it was the most polite forms possible.
Let's play soccer at 1 o'clock
Let's play soccer this weekend
One possibility is that since only 週末 was used, that に would be required to imply "this weekend" instead of "on weekends."