Of course system matters. The real question is how much does system matter?
I've had good times playing games with poor [1] rules, and bad times playing games with good [1] rules.
It seems to me that Traveller is a game in which background (canon) is more important than the rules system.
However rules systems do have some influence. I remember when I was playing in Aramis's Traveller game (almost twenty years ago). We had just switched from TNE back to MT. While the other characters were in a bank, my character, Sigrid, was relaxing on the beach. When the bank was robbed she ran over to help them, and learned how much a quadruple damage shotgun blast [12 point hit in MT terms] can hurt an unarmored character. If she hadn't had high stats, and the damage rolls hadn't been low, and she hadn't known psionic Regeneration, she would have died on the spot. [In retrospect she probably made herself a prime target when she telekinetically 'borrowed' one of the other robbers guns.]
If we'd still been playing TNE she'd have had no chance of dying from a single shotgun hit, and may not have been seriously injured (unless it had been a head hit). System matters. [2]
There are other games (Dungeons & Dragons) in which the feel of the play experience is more important than both the rules system and the background canon.
There are other games (HERO System) in which the rules system is more important than the background or the feel of play, because the rules allow you to play in many different backgrounds in which feel of play may vary, all using the same rules set, with the addition of supplemental rules for specific campaign types (knock-back for super-heroic campaigns, etc.)
[1] Arbitrary and undefined, as this is very much a YMMV situation.
[2] I'm glad we switched. PC toughness in TNE is implausibly high for me. OTOH I'm o.k with D&D characters being that tough because setting matters.