There's a few separate things to consider here. I'll try to cover each succinctly.
Your advisor probably (hopefully?) knows the field well. They should have a good idea of what would and wouldn't be accepted. I'm sure they will have been in a similar situation in the past, and have experience of it. Therefore their view on the matter is perhaps important from the angle of experience and subject knowledge.
It may be that your advisor simply doesn't feel comfortable publishing the work as it stands. As a professor, there is a lot of pressure to publish high quality works. As academia is highly competitive (at least in some disciplines), perhaps your advisor is worried about publishing ongoing work at an early stage for fear of giving a competitor an advantage? This isn't a negative remark as to them, rather them looking out for both your interests (as you wouldn't want someone to snatch this away from under you).
Having a discussion about the situation (as I believe you began) is important in these situations.
Perhaps this is discipline specific, but you could maybe approach the subject by suggesting a recent results track (or ongoing work) in a conference? Often these are good places to sound out ideas and seek feedback or potential input or collaboration. You would definitely want the active support of your advisor for this though, since they would likely be the one paying the conference fee and travel expenses.
It may be that your advisor simply doesn't feel the work/results so far are good enough, or that the experimental procedure (if applicable) was suitable, given the early stages of the work. I know of situations in the past where (for right or wrong, that's a whole separate debate), advisors have told students to remove their name from a submission if they were going to insist in submitting it. While that's hopefully not the case here, it often is an indication of the advisor's feelings on the matter, in them saying they don't wish to be associated with the work.
I would suggest you perhaps approach the subject with your advisor along the lines of point 2 - perhaps suggest a recent results or ongoing result track at a smaller conference? You would hopefully then be able to discuss the advantages or disadvantages of submitting for publication. I can't comment on the offer to do the writing yourself, as that was how I worked anyway, but I'm well aware this is probably disciple specific.
At the end of the day, your advisor will probably make a recommendation as to how to proceed. Remember they likely have a fairly good idea what will and won't be accepted for a conference or journal, and perhaps simply want to save you the time in preparing a manuscript for what they feel is inevitable rejection. So talk to your advisor, and perhaps propose a destination with a suitable recent results or ongoing work track?