Was he supporting his speech? I think he gave a speech but used references and facts to support the assertions or conclusions made in his speech. Since "speaker" is singular, the possessive adjective should also be singular, "his" or "her". It would be unusual to hear this in the present tense unless someone was making the comment to you during the speech, in which case it would probably be in present progressive. i think this would more likely be said in the past tense. It is not just not a matter of personal opinion. The point is that the speaker made it clear to his or her audience that it was not his or her opinion. So here is a possibility: The speaker supported her assertions with references and facts so that no one would think that she was promoting her own opinions.

I think the sentence is right.

"The speaker supported their speech with references and facts so as to not make it a matter of personal opinion" 'Their' is probably more natural than 'the'. I'm guessing that past tense is more appropriate? You need something like 'so as to' in there to link it. You could also say 'to make it less of a matter of personal opinion'.