Depends. The SPG has greater range, so may be able to affect an area without having to get as close, meaning it may not have to access terrain that the M777 would have to. DSTO have done studies that show the extra range a SPG brings over a towed gun means it can hit any feasible target a towed gun can without having to go into unacceptable terrian. As for lightness, the M777 would be the lightest towed 155 mm - it's only 4.2 tonnes, which honestly isn't that heavy (it's 2/3 the GVM for a 6x6 G-Wagon...). Certainly the AH4 (at 4.5 tonnes) is sold as lightweight as it's 25% a normal 155 mm system.
And that has two problems. One is there are compromises within the M777 design that means a M198 is tougher and more forgiving if you want to use it in an off-road towed role. As with anything, you can't throw something around in our outback unless it's pretty tough, and that toughness comes at the expense of weight.
The second problem is that a CH-47 only lifts one gun + crew. The ammo comes in a second helicopter. So each 4 gun Bty needs 8x CH-47 lifts. That's 75% of our fleet. It's going to be unlikely that such numbers are available - it's an enormous commitment. When we start adding in times - about 10 min from gun unhooking to first round downrange, an anemic rate of fire that puts 2 - 3 rounds down range per minute (closer to 1 - 2 in mountainous terrain) and you've fired 40 rounds in 15 min. But Russian counter battery fire has an open source time of 80 s in the Ukraine. Assume 50% worse, and your helicopter inserted Bty has fired 16ish rounds before getting hit.
I was taught helo-bourne artillery raids many years ago, and as I've moved through my career that's fallen off as no longer realistic. We cannot guarantee air superiority, the guns fire too slow and we need to commit too many of our most valuable tactical transport. While I get annoyed at people who say Tiger replaces artillery (and that was a common view in the 2000s), the reality is what you threw an arty raid at 25 years ago you would now through a Tiger Tp at. You'd be faster, more flexible, more survivable and less resource intensive (even from a $$ point of view!)
Do we? When you say logistics train I assume you mean spares for the gun system - and you'd be correct. If you mean for 155 mm rounds, no, we don't. Very few people have an idea on how much 155 mm is chewed up by an artillery Bty or Regt when firing for real. We have a dedicated Transport Sqn for this reason - but poor exercising of artillery over the past decades has seen that Sqn progressively directed elsewhere.
There is no room for towed guns in modern warfare. Unless you have a huge helicopter fleet, huge budget (so your M777 or AH4 only do helo stuff) and you don't really care about other capabilities. A SPH is better than a M777 in almost every respect at every level (tactical, operational, strategic). Those it isn't, it's equal. The one niche capability of airlift demands more online CH-47 than we have - and hence is not cost effective. The M777 capability (because it's more than just the gun) is slower, clumsier, longer, less agile, more manpower intensive, less safer and less capable. For every $ spent on an obsolete system like the M777 you lose a $ that could go to the SPG. There is 0 reasons for the ADF to keep towed guns, especially with the opportunity to purchase increased numbers of SPH.