This is a question well suited for a non-linguist. I have wondered about exactly this for the past few years and have come up with a system which is almost finished. I think the way to look at it is there are about 24 clearly different consonant sounds, and 12 different vowel sounds (including the English "r" as a vowel which I think is better). All the other consonant and vowel sounds are variations on these base 36 sounds.
If you are not an experienced native speaker or linguist in a particular language, and you were to be able to pronounce the 36 base sounds, you could get by as a beginner. Then as you learn how to get more refined and produce the variations of the sounds, you can move out of the beginner stage into the native stage.
What are these 36 sounds? I have started to map them out here. Letters without diacritics are the base sounds. Still working on how to present this so take with a grain of salt.
But here is a map of some of the initial values mapped to IPA.
In the second column are what I call the OVO script, a latin script with minimal diacritics to get it across. OVO meaning "One Voice Orthography". The letter forms you see in the image above are a more expansive set of characters (36 shapes) to account for all the base sounds without diacritics. Again, it's in the final stages so not 100% exact, but if you click the "find word" links for various languages, there is an initial attempt at collecting some words in Arabic, Tibetan, Hindi, and others using this lettering system, which I am calling Hanakana.
So in short, yes, I believe there is a simpler way to deal with the IPA system, that will get you to pronounce any language with at least a beginner level of skill. Then adding diacritics (dots and arcs) to Hanakana, you can capture almost all of the IPA sounds. By almost all, I mean capturing most of the variations without getting into an exact replica of IPA. Because even IPA isn't 100% accurate, it is a set of approximations of the vocal chords and shapes of the mouth and air moving through these crevices as you've probably seen. So with this system here, you can do everything you would need to to speak languages from humans on earth, with current human anatomy.
To account for aspiration, add an "h" equivalent after the character, like normal. To account for nasal vowels, add a dot below, etc.