The Romans certainly did have spelling and spaces. Without those things you don't have words. What they didn't have was punctuation and syntax. Grammar in Latin is built into the words themselves, via conjugation of verbs and declension of nouns, with proximity determining what word modified what.
Your point still stands, though. If you don't bother to spell properly then you just look like an idiot.
Im certainly no expert on the subject. Also, it was somewhat of a dissapointentment to realise in later life that a lot of stuff teachers in school taught us was simply wrong. I hope this is not one of those things, but you never know.
Anyway, I was taught that on the clay tablets the Romans used for day-to-day things they did not use spaces, this to conserve writing space on the tablet. I was taught that this made an archeologists job more difficult, as they had to figure out where words ended and began, since romans also solely used capital letters.
Of course, you are quite correct on what you point out on grammar.
I have read that if mankind ever were to meet aliens, a likely language to first try to communicate in would probably be Latin, as Latin is very close to math in regard to how it is structured. Logical, predictable.
Unlike for instance English, which is sort of a hotch-potch of languages and where spelling is mostly convention.
So the Roman example is wrong, but still, Shakespeare isnt even consistent in how he spells his own name. The convention of spelling does serve a purpose.
Terry Pratchett once wrote English is the sort of language that mugs other languages in a dark ally, and I thought that was a funny way to put it.
My own spelling can hardly be called perfect, I use what I call the ballistic approach. Meaning I sort of point at my goal, fire, and hope for the best.
But at least I make an effort and I am always happy to learn new things.
(Twenty years of using English and only three months ago I find out that the only correct spelling is 'which', Id been writing 'wich' for years..)