Wintec makes some very top-dollar(and some cheaper models as well) synthetic endurance ,(not sure about western) saddles that are very comfortable(if you want to spend anywhere from 1-2k for them!), so synthetic saddles can be very nice. However, the cheaper brands in the $100-$200 range often can have bad stitching or poor quality synthetics, so I'd perhaps ask for reviews on FB or a group of people who've actually owned and ridden in them for a while before buying one.
Anywhere you go in the show world, people are going to tell you that you need to get fancy tack, fancy clothing, etc. It's just how it works. I went through a "rodeo phase" as a teen but when I saw the costs of saddles, tack, boots, etc, I quickly got out of that. I used to show 4-H and open class shows as well when I was younger and the amount of money some things cost was utterly ridiculous. However, I did find that you can often find some pretty decent quality things used or older and they work just fine. I have an old barrel saddle I got for free that looks gorgeous still and works just fine for me as a regular trail saddle(it's an older model, so no 4-inch horn lol). My black show saddle was $250 and is full leather, with barely any use. I don't see the need to buy new tack unless I absolutely cannot find something good quality used.
My advice? If you can, buy it used. Leather tack that's a bit more plain and less-bling is often pretty cheap and you can always embellish it a little on your own if you'd like. Personally, I prefer a more classy and simple look better than the bling anyhow. (You won't catch me wearing crystals or a bunch of silver...) If you ask around and find good reviews for the synthetic saddles outside of the website or Amazon reveiws/etc, go for it! There's no reason why you need to spend a ton of money on a saddle when just getting into a sport that you might not even want to stay in after a while of trying it out, and absolutely no reason why a nice synthetic saddle shouldn't work just fine.
"The art of riding is hard to find and easily lost."