From the yellow paper:
2. The Blockchain Paradigm
[...] Blocks function as a journal, recording a series of transactions together with the previous block and an identifier for the final state (though do not store the final state itself—that would be far too big). They also punctuate the transaction series with incentives for nodes to mine. This incentivization takes place as a state-transition function, adding value to a nominated account.
11.5. Mining Proof-of-Work.
[...] However, since mining new blocks comes with an attached reward, the proof-of-work not only functions as a method of securing confidence that the blockchain will remain canonical into the future, but also as a wealth distribution mechanism.
This answers your main question. Why not just let the reward just be the fees of the transactions included in the block? Maybe it wouldn't be a good enough incentive.
"Why doesn't Ethereum have a higher limit? Will it ever exist? If so, why? Wouldn't uncapped coins have an inflation effect?"
Uncapped coins do have an inflationary effect. Still, remember that there isn't everything set in stone about this. There may be changes in the future and a lot of factors must be taken into consideration by the developer team.
As it is said in this article:
Ethereum, or rather Ether, the cryptocurrency of Ethereum, does not have a deflatory nature, but an inflationary one. In short, you will never stop issuing new ETHs. According to some forecasts, the inflation rate of the circulating supply of Ether will tend to approach a rate of 1% by the 64th year of its existence, that is in about 2078.
However, this does not mean that rewards will not decrease.
Every now and then they rewards are reduced, as the growth graph of ETH’s circulating supply shows, even though this is not predetermined, and we do not know of how much premiums will be reduced every time. In Bitcoin the problem does not arise, because halving is predetermined and is an integral part of the protocol, while for Ether the decisions of when to reduce the premium, and how much to reduce it, is made over time.