given that the distinction is not made in gaelic.
Why would you say that? There is the distinction. Tha sinn a’ cluiche… is we are playing, at the moment.
We play (once in a week, sometimes, habitually) would need the future tense (which, besides future, has also the present habitual meaning): bidh sinn a’ cluiche or cluichidh sinn (the first roughly equivalent to the Irish present habitual construction: bímíd ag imirt or in Hiberno-English we do be playing, the latter to Irish present tense imrímid).
I believe bidh sinn a’ cluiche would be more common, but note examples from taic.me.uk: sgrìobhaidh sinn litir thuige a h-uile là we write (to) him a letter every day, gabhaidh e bracaist a h-uile madainn he takes breakfast every morning.
Scottish merged its older present and future tenses into one ‘future’ or maybe better ‘non-past’ tense that also carry the habitual meaning. Both Sc. Gaelic and Irish use the continuous verbal noun construction for the progressive at-the-moment meaning.
(EDIT: seems I had a misconception about a habitual present in earlier language, edited it out)
Because, as I tried to explain in the other comment above, this sentence doesn’t translate to we play. It’s we are (right now, in this very moment) playing together.
Also not sure if together with each other would be accepted (it might, I have no idea if the contributors anticipated it or not), very literally (and certainly not accepted) though it’d be something like co-hand to each other; hand-in-hand to each-other.