TL;DR: when you see a plain "C" chord symbol, having C as the bass note is the only safe option that's not wrong. (Except if there are instructions to not do that, or if you know that the written chords are wrong to begin with)
Musical notation, including chord symbols, is written communication from people to people, happening in a cultural context. It is subject to the same phenomena, deficiencies, mis-interpretation, guessing etc. as any other kinds of written communication from people to people. The writer writes something in order to communicate ideas and to cause some actions to happen with the readers.
What do writers expect to happen if they write a "C" chord symbol? Most of the time in simple pop songs they want to tell the reader that a C major chord with C as the bass note should be played, at least on a rhythmically strong moment. I don't think music writers are offended if someone plays something slightly different, or plays a walking bass line or something, as long as it's done with good taste and without putting the composer or arranger to shame. If a C/E sounds good there and serves your musical purpose, go for it. It's easier to just do it and apologize later than ask for permission...
However, if the chord symbols are meant to be used as a strict harmonic analysis, then it means the notes C, E and G and C being the lowest note. Whether it's a practical lead sheet for a pop song, or a theoretical harmonic analysis - I hope people can tell the difference!
What do readers expect from chord symbols? I think that there's an implied promise, "if you act according to these instructions, then it will sound at least roughly like the song you expect to hear." I guess many people have been disappointed by the approximations in pop song books not really fulfilling that implied promise. If you look at transcriptions of pop songs, there might be lots of sounding notes on a definitive popular recording in addition to C, E and G - or even completely different chords - but still it's just written as "C". Why? Maybe for the sake of simplicity, or the transcriber's decision that extra notes aren't really relevant. Or perhaps because the transcriber was in a hurry or lazy, or wanted to get credits and money for making an "arrangement". ;) Then someone looks at the transcription and complains about it being "inaccurate". Sure it's inaccurate, it's always a compromise. So, if you see just "C", but you know it's wrong and your audience expects a C/E, then play C/E! ;)
What comes to guitarists - if there's a bass player who's playing lower than you, then by all means play any kind of inversions you want, as long as you don't make a low-frequency mess by stepping on the bassist's toes. It may make the sound more open if the bass notes aren't always doubled. And it might even be better for guitarists to avoid low notes completely, to avoid clashing with a bass line, if the bass player likes to improvise and if the bass isn't 100% tightly coordinated between guitar and bass.
A lot of guitar driven sites portray chords only in root position,
almost making it 'wrong' to do anything else,
This is a good point. I think it would be great to remind guitarists that they do not have to play all the low notes indicated in the fingerings. It might make the chords sound more interesting if you leave out the lowest notes and let the bass player handle the bass.