What purpose would this serve that the Aramaic Alphabet doesn't already solve?
That isn't the point. Latin just seems more convenient in this day and age. So it would seem appropriate to use it.
Secondly, the Latin Alphabet cannot support our various sounds that Aramaic already covers.
What sounds exactly? The guttural "kh" is supported by that digraph. The emphatic T sound (as heard in Arabic "tamata" = tomato) is supported by a diacritic, or an accent below T. Latin has always been open to such alternatives, whereas other scripts like Cyrillic and Greek aren't. That's why most of the world uses Latin, from Turkey to Indonesia. Latin can easily incorporate "foreign" sounds.
How about teaching the kids the language and alphabet before they learn latin and english?
I grew up using Latin but that didn't stop me from reading the Sureth alphabet.
Well, this doesn't seem like it's working. You and I are interesting in Syriac alphabet, but many aren't unfortunately.
Thirdly, the Latin alphabet doesn't sound our similar vowel sounds. How would u represent PtaHa and Zlama in Latin?
What's wrong with our A, E, I, U and O's? They have always represented the Syriac vowels. Again, long vowel sounds can be represented by accents or diacritics above or below a Latin letter. Most European languages have this feature.
Also Latin isn't older than Syriac. Syriac is a derivation of the Aramaic alphabet which was used to form the Greek alphabet which spread and eventually formed into Latin.
The Syriac script came to be in 200BC after evolving from Aramaic (800BC), which developed from Phoenician (1200BC). The Latin alphabet appeared in 600-700BC, and it was rather close to its "mother" scripts, Etruscan (700BC) and Greek (800BC), which evolved from Phoenician. As you can clearly see here, Aramaic and Greek are the "offspring" of Phoenician, and have spawned their own derivatives or ascendants.
You can say that the Greek script is to Phoenician, like how Syriac is to Aramaic. Now that doesn't mean Greek is over 2000 years old just because its parent script is that age. As such, it's not logical to say that Syriac is "older" than Latin just because Aramaic is.
By the way, Latin still has letters that look similar to Phoenician, whereas Syriac doesn't, despite sharing the same alphabetic names and being an abjad (like Phoenician). A lot of the Latin letters still retain their "ancient" look, whereas Syriac looks almost completely disparate in contrast to Aramaic and Phoenician. Even the Hebrew script barely looks like Phoenician. Funnily, the Aramaic letters look more closer to Latin and Greek, than they do to Syriac and Hebrew. Take a look at Aramaic's aleph and beth, and compare them to Latin's A and B.
I find it strange how Syriac letters evolved to look so much different from Aramaic's letters, whereas Latin and Greek still have a resemblance to some of the Aramaic and Phoenician letters. And even now Syriac has a two-script system that don't look too homogeneous (madnkhaya & estrangela).