Word is a popular term, which means that everybody has their own idea about what it means, aided or hobbled by whatever education they may have encountered that claimed to deal with language or languages. That varies quite a lot.
In other words (sorry), linguists don't define the notion of "word", except in particular circumstances, under particular conditions, for particular purposes. It's not up for general revision, and linguists have no authority to change people's vocabulary in any case.
Linguists only get to define technical linguistic terms in a technical linguistic context, like, for example,the fact that there are only two tenses in English, instead of the 6 or 12 or 67 that one might have been told about in what we laughingly call "grammar school". Or like Matthews' distinctions between three different senses of word.
English is actually a pretty analytic language, which means that there isn't much inflection, so mostly syntax is a matter of the proper choice of lexical items and their arrangement into constructions. Word is a useful choice for these lexical items, as long as we don't get bogged down in pointless determinism like the question of whether
XYZ is really "a word" or not.
In a polysynthetic language like Lushootseed, on the other hand, there are a lot of situations where it's hard to tell the difference between a word and a sentence.
- háʔɬtәƛʼusyiqʼibʔәcәluƛʼ 'The old woman used to make good baskets'
has three semantic roots:
1. haʔɬ '(be) good, well'
2. yiqʼ- 'weave [baskets]'
3. luƛʼ 'elder [person]'
Everything else is supplied by the multiple inflections or derivations, rather than by auxiliaries and articles, prepositions and particles, like English.
And even in English it's not so simple to identify words, let alone count them.