It is possible to create a system that doesn't use its own token like ETH, and instead pays miners in some other asset like BTC, USD or chickens. But it is easier to secure a system if it has its own token. A longer explanation follows.
As you say, miners are paid in ETH. They are paid in two ways: Through the mining reward, which gives them a fixed amount for each block they mine regardless of what transactions it contains, and through fees, which are paid by people making transactions.
In the future, it is planned that Ethereum will change to Proof of Stake; When this happens the nature of mining will change but there will still be a need for people to perform a similar role to the role that miners play: Gather transactions into timestamped blocks, and add those blocks to the existing chain of blocks, providing evidence about which of various possible chains of blocks will be considered the canonical one.
Either way, the people making blocks need to be paid.
One benefit of using a native token is that it allows the protocol to create new money to pay for the block reward. This is unnecessary if you design a system that doesn't require a block reward; Public blockchains can probably be operated without one, and funded purely through fees. The design of Bitcoin calls for the block reward to eventually stop, at which point it will be funded entirely through fees. This may work, but it is largely untested, and has some theoretical problems; For example, there is a concern that miners may not have an incentive to keep building on other miners' blocks, making a longer chain, when they could instead ignore those blocks, and try collect all the fees in them for their own competing block. So it seems easier to make the system secure if you have the ability to pay a block reward. If you don't have a native token, that money has to come from somewhere, so you need to somehow find a way to pay for it.
The other benefit of using a native token is that it helps align the incentives of defenders and attackers. This is particularly true in a Proof of Stake system. If the system becomes very valuable, the price of ETH goes up, so the value of the block reward and the fees, and thus the amount of resources put into defending it, also goes up. In a proof-of-stake system, this also makes the amount that would be lost by breaking the system go up.
PS. As a general-purpose network, Ethereum has a security disadvantage that a system like Bitcoin, which is almost entirely used for managing its own token, doesn't have: Although Ethereum has a native token (ETH), much of the value on the system is not in ETH: It's in other tokens on the network, or other data and applications. Generally it's assumed that as these become popular ETH will also go up, but in theory it's possible that in future, even though the Ethereum network is very useful and manages very valuable assets, the actual value of the token will be much smaller. This may make it attractive to spend money attacking the system, in the hope of stealing some asset that it manages, which is more valuable than the token that is used to pay for securing it.
