Even-Mansour is not as severely limited as you make it sound. The success probability for an attacker of Even-Mansour has been shown in various papers to be at best
$$
\text{Adv}^{SPRP}_{EM_{K_1, K_2}}(q, p) \le \frac{2qp}{2^n}\,,
$$
where $q$ is the amount of black-box queries you make to the cipher (read: known or chosen plaintexts), $p$ the number of queries you make to the permutation (read: bruteforce), and $n$ the bit width of the permutation. The same security bound applies to the single-key variant
$$
\text{SEK}_K(x) = P(x \oplus K) \oplus K.
$$
Now, this means that, unlike an optimal block cipher, in which no matter how much data you collect you still expect around $2^n$ computational effort to break it, here by collecting $q$ plaintexts the effort goes down to $2^n/q$.
Now, to answer your questions:
Even-Mansour is already secure, as long as you adequately deal with the above caveats. For example, if you choose your permutation to be, say, 512 bits wide, you have nothing to fear from any realistic attacker.
Yes, iterating several permutations does give you better security. For example, the key-alternating cipher (or iterated Even-Mansour)
$$
\text{KAC}_{K_0, K_1, K_2,\dots,K_t}(x) = P_t(P_1(x \oplus K_0) \oplus K_1) \dots) \oplus K_t
$$
achieves better security $$\text{Adv}^{SPRP}_{\text{KAC}_{K_0, K_1, K_2,\dots,K_t}}(q,p_1,p_2,\dots,p_t) \le \frac{4^tqp_1p_2\dots p_t}{2^{nt}}\,,
$$
which quickly converges to optimal as $t$ grows.
As seen above, iterating Even-Mansour is an effective way to strengthen the cipher. In practice, however, having several independent permutations and keys is inconvenient, and so you see designs that use a single permutation along with multiple round keys, each derived from a single master key. The AES is one such design, with the public permutation—the round function—being weaker than a random permutation but efficient to compute. This is a very successful design principle, and you will find many other ciphers following the same strategy.
