Would NSNA been a bigger hit if the action scenes were higher budget?

It seems to me, that's the only problem with the movie are the action scenes. The story is just as good as any other Bond movie that was out at the time, and the villains were overall solid I thought. I also thought that Connery's performance was good too overall, just as good as Moore's at the time. So it seems that the only thing wrong with this movie is the lack of budget on the action scenes, but was their more wrong with this movie than just that?

Doubtful. There seems to be a distinct lack of energy combined with some over the top acting and dialogue. Combine that with some truly awful scene setups/mechanics, mediocre score and lack of the Bond tropes, people would have a hard time buying into it being a Bond film.

I can't say it would. I loathe NSNA on a good day but removing that from analysis, I think going up against an actual Bond film doomed NSNA. Remaking a film that some say was slow, sluggish and overbloated can't have settled in somebody's mind well when going to watch it.

Oh okay, I just thought that if the budget was higher, it may feel and look more slick and Bond fans would therefore be more accepting of it.

It's the scripting issues, the production problems, the legal battles that were ongoing and the constant pulling and tugging in different directions form all over the place. The producer basically disappeared on set and it was his first feature. It was a very difficult and unhappy set and barely managed to get to the finish line.
The task largely fell to Kershner and Connery to get the darn thing finished and it is their efforts that resulted in a finished picture at all. They came up with great bits and the good parts of the scripts that still work are the best parts of the film.
Everyone in the production should have been entirely on the same page and they weren't and it kills the film's effectiveness. What works are the riffs on the official series, the sense of fun sprinkled throughout, the witticisms of the different screenplays that make it through and both Connery and Brandauer's performances which are sublime. It's the definition of a wasted opportunity and mixed bag.

Even the look of the film is quite flat which is surprising as it was shot by Douglas Slocombe post Raiders. The pacing is dead in the water, the tone is too relaxed, there is an almost TV movie feel throughout, the flat contrasty look of video editions does not help, many of the performances are off the mark or underwhelming etc. Kim Basinger is useless as Domino and literally has one facial expression for the whole film.
The action actually is one thing that works fine. Nothing special and the underwater finale is beyond uninspired but the fight in Shurblands is overepic, the opening is ok if a bit cheesy, and the shark attack is inspired and all done for real.

The best scene in the whole film is the Domination game. The best official series joke is the new Q. The glorious golden gem of the film is Nigel Small Fawcett.

Don't get me wrong I've come to appreciate the film's strengths over the years and watch it after my series re-watch at least once a year with CR 67. But the more you read on the production history it was just impossible to pull together. We're lucky we got anything at all.