Now here's an interesting theme.
I very much doubt about Palmyran's abilities.
I have only lost against Palmyran when they were handled by players that were way faster than me - the Player, not the Civ, made the difference! If they were playing Greek, they would still have beaten me. palmy had nothing to do with it.
Now to useable comparisons (good players, but not significantly faster than me).
I have beaten them with:
Roman:
Bronze:
My +4/+4 Cavs and some Impies along with a few ST reduced Palmy's Camels to pulp.
I used ST + Towers + some Cavs to reduce and even erase the merits of their best Bronze combo (Composites + Stone Throwers)
Iron:
My 12-range Helepolis solved the matter - always. Roman Scythes are superior, too, but you won't be able to cope with the ease with which a competent Palmyran can spit out Scythes and HA's. Helepolis do the job nice.
Minoan:
A variant of the Roman, but with a stronger Bronze.
In Iron, Helepolis again...
Hittite:
Superior Catapults and better Scythes do it.
If the Palmyran go Armored Elephants the Hittite has more trouble in Iron, but a few AE's of the Hittite coupled with a backing by HHA - OR - EA in front of the Cats (especially Heavy) can do it).
Other Civs:
Don't know, but Sumerian should have no problem with them.
Shang, of course, can outproduce and outgun them, because of shnag's competence in flooding.
Shang fears no Iron, the Heavy cat + helepolis combo, even with less range is very powerful - especially preceded by hordes of Scythes. After the Scythes die, not much will be left to deal with the approaching Siege.
Shang is MURDER.
So, IMHO, Palmyran has more problems than anybody else.
The advantages of Palmyran are yet to be materialised, in the sense that they simply can be slaughtered in Tool, their Bronze is Ok, but other Civs will compensate by having MORE Villagers... that's more FLEXIBILITY (more choices for simultaneous gathering of different resources), more SPREAD (=safety) and more building power (I very often don't do military UNTIL I know what I will be attacked with - at that time it is handy to produce a mass of what is needed and nothing else This is highly efficient when you cannot wall in.As for Iron, the lack of key technologies very much impairs their firepower:
Engineering, Helepolis, Craftsmanship, Metalurgy are in fact decisive, especially when they are all lacking. Less range for siege and less range for missile units really hurts - also with triremes and towers. Scythes also do less damage, so they cause less problem before they die.
As it has been said before, any player winning with Palmyran... could probably win with another race.
Having it all, but 2nd. class, is not very attractive.
Shang would fall here, apparently - but the fact is that they are first class in the economic department, so they have first class flooding!...