Shalom. I love that you asked this question as it is a very important issue.
Many people in Israel do make these distinctions an so do not suffer from this ambiguity. We now know for a linguistic fact what the 32 sounds are the consonants make. Now to answer your question. What reaction would an Israeli have if you spoke unambiguous Hebrew? Well from my experience in Hebraic circles some would like you for it but others will also hate you bitterly for it. Especially the 'w' sound. I have been kicked out of groups just for using the 'w 'sound in transliteration and told anything other than sephardic or ashkenazic is not tolerated. Because of this I have created a Facebook page called "learn Hebrew uncensored" for those that love unambiguous Hebrew. Most unambiguous Hebrew is the same but one letter called 7ayin can be ع or NG like siNg. Either is fine. Unambiguous is what matters. It's estimated that just adopting two sounds in Hebrew can reduce the memory load by 20%! Furthermore ambiguous Hebrew forces the learner to look up words in the dictionary 5 different ways. It wastes time.
To quote a mizrachi in Israel, "the Ashkenazi accent flattens Hebrew, narrows it and, worst of all, makes it confusing and vague" Says Tsadok. "Hebrew has become a crippled language that needs crutches to be understood by its own speakers."
Here is a YouTube video explaining the transliteration numbers in Mnemonic Hebrew
What are those numbers in Mnemonic Hebrew?: http://youtu.be/UjQzmIl6CX8
Also search learn Hebrew uncensored on Facebook. The one place that supports and tries to preserve native sounds of Hebrew.