Well I have to say I sort of disagree. First no one should use his starting 1000 to get the hp to towers upgrade (600g i think).
Instead he should get an item that will help him stay out longer so as to creep and earn gold/exp. The towers aren't going to be destroyed within the first 6 levels unless there's an unharassed rook. Damage from creeps isnt significant either. Demogod's usually prefer avoiding attacking towers due to loss of hp. They will never attack a tower so early if you have your demogod there.
But the main reason to not get a hp+ to towers is because the main defender is your demogod and not a pile of bricks. Your demogod should be preventing your opponents from gaining any foothold and being able to harass your tower. If due to being physical(buying cit upgrades when the other team buys hps+ items) weaker than the other demogod you forced to retreat, that's a loss (just a small loss, no reason to rage quit) and already the other demogod has time to creep while you hurry back to heal up.
anyway importance of tower upgrades:
1. HPS to towers 1
2. Gold income 1
3. During this period you buff your demogod equipment.
4. Purchase priests only as late as possible.
5. angels + cataphets should be purchased at the same time and near the end of the game or for the enemy portal flag capture strat.
6. giants to finish the game
7. everything else is rather not important , well perhaps the splash damage is an alright upgrade as well but not required.
Anyway last weeks game I played where a team went all items and no cit upgrade at all (except for gold), while the other team went lots of citidal upgrades. It ended up with one team at 17-18 while the one with cit upgrades was at 13 at game end. And those level 13's were only a little better than the ai giants.