That's a lot like asking what weak points dislikes are there with Remington centrefire rifles. Are you talking an SPS? Sendaro? 710? 7400? Did you have a specific Savage in mind or are you just looking for comments all over the map much like Savage's line up of rifles is?
Every single firearm design... EVERY ONE...has weak points. Some more than others, but don't kid yourself, they ALL have something that isn't perfect.
I prefer knowing what those weak points are, before I commit to a design.
Its a bit like dating. Sure, we mostly do it for fun...but at a certain point we are making the occasional mental note about whether this chick is going to be... a nighmare to actually live with long term... a poor mother, or a great mother... gold digger looking to clean you out... "date" the town while you pay the rent... or be faithful and trustworthy. Now, you can never truly be 100% certain, but if you are honest with yourself you can decide which of her flaws you can tolerate or overlook, and those that chill your spine and think "I gotta lose her number asap!"
Or put it in car terms: you wanna have some idea if the car you are buying is undergoing a serious recall, has a tendency to catch fire seemingly randomly (do a google search of pics of supercars on fire!), or whenther a previous owner lost his license for infractions... or if they have meticulous maintenance records (not likely the same person). You wanna have some idea of what kind of baggage you'll have to deal with. Or if its an older muscle car, you wanna know if there is a casting flaw in your engine block that keeps oil from flowing at high volume when doing the 1320. Learn about the flaw, take steps to work around the flaw, rendering the problem neutralised.
The accuracy of the Savage rifles is appealing to me. The ability to swap barrels and calibers at home or at the range, its almost Lego modularity is very attractive. But... if it falls apart, or is unreliable at times, I want to know details: lack of cleaning? Old worn part? Design flaw specific to specific conditions? Are these failures something I can prevent by taking care and attention, maybe planning ahead... or is it a "Hangar queen" requiring more effort than its worth?
To be more clear, the budget Axis line isn't what interests me, I'm thinking the higher grade units. But...the Axis is still of interest to this discussion, indeed all the short and long actions, because its basically the same action. Flaws appearing in one, may appear in them all, potentially. Sure, mag differences, different stocks, different lugs mean there is some variation, but I'm looking for people's real world observations.
If I had any interest in Remingtons or Weatherby's I'd be asking the same about those. But I'm not. We learn more from examining mistakes and flaws, than from perfection.