OP asked:
What reasons are there for believing that the words of the historical Buddha are preserved?
What reasons are there for not believing that the words of the historical Buddha are preserved?
Do these reasons apply equally to all "words" or, for example, is it more believable of the suttas than of the Jataka tales, or of some suttas more than others?
The first good news is that a whole book has been written to answer these questions.
It's The Authenticity of the Early Buddhist Texts by Venerables Ajahn Sujato and Ajahn Brahmali.
The second good news is that the whole book is available for download, for free, in the PDF format.
The Abstract states:
This work articulates and defends a single thesis: that the Early Buddhist Texts originated in the lifetime of the Buddha or a little later, because they were, in the main, spoken by the Buddha and his contemporary disciples. This is the most simple, natural, and reasonable explanation for the evidence.
Our argument covers two main areas:
The grounds for distinguishing the Early Buddhist Texts (EBTs) from later Buddhist literature;
The evidence that the EBTs stem from close to the Buddha’s lifetime, and that they were generally spoken by the historical Buddha.
Most academic scholars of Early Buddhism cautiously affirm that it is possible that the EBTs contain some authentic sayings of the Buddha. We contend that this drastically understates the evidence. A sympathetic assessment of relevant evidence shows that it is very likely that the bulk of the sayings in the EBTs that are attributed to the Buddha were actually spoken by him. It is very unlikely that most of these sayings are inauthentic.