The tricky part with modifying a decentralised system, like Ethereum, is that you need to incentivise both the miners and the users to want to keep upgrading (to a Proof of Stake consensus mechanism) - to avoid "protocol stagnation".
There are lots of references to the time bomb in Ethereum which ... is meant to control inflation of the currency, because it is growing too fast right now.
I'm not sure where you got this information about inflation happening too fast, but I disagree with your statement.
Based on Vitalik's comments in past developer all-hands calls (such as Meeting 14) where he wasn't concerned about pushing back the Ice Age despite the effect being that the market cap of Ethereum growing by 2-3 million Ether - saying it wasn't a big deal.
Increasing difficulty will also increase the block time, which is not good for transactions.
That's correct, it's not good for applications that use Ethereum - this is the users' incentive to move over to Casper / PoS; a more usable system.
Why aren't the developers of Ethereum going this way? Is there some technical limitation?
No technical limitation, it's just important to find the correct balance and come to an agreement amongst the community.
An agreement has now been reached and they are lowering the reward to 3 Eth in the next release (which all going to plan with happen on 17th October) - you can see this change in the code:
frontierBlockReward *big.Int = big.NewInt(5e+18) // Block reward in wei for successfully mining a block
byzantiumBlockReward *big.Int = big.NewInt(3e+18) // Block reward in wei for successfully mining a block upward from Byzantium
maxUncles = 2 // Maximum number of uncles allowed in a single block
Ethereum's Difficulty Timebomb / Ice Age, and the recent changes that are coming in the new code release, is actually designed to keep Ethereum progressing forward by incentivising the users and the miners to upgrade; modifying the mining reward alone wouldn't achieve this goal