So was it Rugby Union's influence that would have League players banned from swimming & boxing or was it just the general principle of amateurism of the time?
Reading the papers most of the other amateur sports leaders were saying what the RU did within their own sport was up to them, and it was of no relevance to other sports - but the Association's rules were ambiguous, intrepretations voiced & pushed by men such as E.S. Marks that put the fear of dire consequences into amateurs who thought of playing RL, and they (all the sports affiliated) thus felt duty bound to recognise the RU's declaration, that RL players were professionals, and thus impose the ban across all the amateur sports.
There was a rule in amateur sports that if you played with or against a pro (whether you knew he was a pro or not), you automatically became a pro yourself. This rule was unclear if it applied to team sports, and fear of its consequences stretched very deeply in the minds of many sportsmen - so many didn't risk it by taking up RL. In Brisbane the RL code didn't begin as the QRL, but as the carefully named & constituted "Qld Amateur Rugby Association" - all with the purpose of preventing what had happened in Sydney (plus there was no money in RL in Qld to pay players for first years anyway!)
It was still festering in 1909 when Snowy Baker came back from England - he had taken up pro boxing, but being a former Wallaby, began playing club RU in Sydney - when the obvious was pointed out, the NSWRU argued that being declared a pro in a non-team sport (boxing), didn't make one a pro in a team sport like rugby (a complete reverse of the interpretation of the same rule they banned RL players across all sports with in early 1908!).
It was all eventually agreed that as cricket & golf bodies allowed pros/amateurs to play side by side under the amateur Association, then what the NSWRU decided about professionals was of no consequence to any other sport.