There are a few key methods I know to produce varieties with bad reduction on a specified set, most of which were mentioned already in the comments:

**Twists**: Given a variety $X$ with an action of a group $G$, we can produce twists of $X$ over $\mathbb Q$ from elements of $H^1(\mathbb Q, G)$. The variety will have good reduction at every place where $X$ has good reduction and the Galois cohomology class is unramified, and it is often possible to prove a converse and/or to fine-tune the set of places of bad reduction of the twist.

Daniel Loughran's example of plane conics fits into this mold, taking $X =\mathbb P^1$ and $G = PGL_2$. So does Ariyan Javanpeykar's example of the elliptic curves $y^2= x^3+1$, taking $X$ to be the elliptic curve $y^2=x^3+1$ and $G$ to be the order $6$ group of automorphisms. The same is more obviously true for any quadratic twist family of elliptic curves.

But the fact that an automorphism of order two is sufficient to get bad reduction t an arbitrary set of new places in the case of elliptic curves suggests this is a quite general phenomenon - many varieties have automorphisms of order two.

Outside of some explicit cases like plane conics, the main way to prove that varieties constructed this way have bad reduction is the bad reduction of their cohomology. To make this work, say in the case of an order two automorphism, it is only necessary that this automorphism act nontrivially on the cohomology. You can find many examples here from sufficiently general hypersurfaces, or maybe hypersurfaces of bidegree $(2,d)$ in $\mathbb P^1 \times \mathbb P^n$, but you will always have some starting set of bad primes to work with that may be hard to control. However, the sufficient generality will easily let you handle the strongest irreducibility condition and the "far from curves and abelian varieties" condition. This was discussed by Ariyan Javanpeykar in his second comment.

**Explicit equations**: For a hypersurface or other variety defined by explicit equations, one can explicitly calculate the primes where the hypersurface is singular. If the number of terms in the equation is small, it is often possible to reverse-engineer to get a hypersurface singular at a proscribed set, as for instance is the case with the Fermat curve $x^n+y^n=z^n$, singular at primes dividing $n$.

However, it is not obvious that a hypersurface has bad reduction at the primes it is singular - for instance, the Fermat curve is actually nonsingular when $n=2$, but the related $x^2+y^2+z^2=0$ is not. Again, the most convenient way to rule this out is to use the explicit cohomology.

However, hypersurfaces that are simple enoguh that their singular primes and cohomology can be easily calculated will not usually have motive independent from curves and abelian varieties, and will almost never satisfy your strong irreducibility condition. To avoid the curves and abelian varieties thing, a good idea might be to use the Dwork family or similar hypersurfaces which have hypergeometric Galois representations appearing in their cohomology, and use irreducibility properties of these hypergeometric representations, but this will certainly not make the whole Galois representation irreducible. Still, these will certainly be irreducible in the sense of not being a product of lower-dimensional varieties.

**Shimura varieties and automorphic forms**: Shimura varieties (for instance the powers of the universal families of elliptic curves with level $N$ structure) usually have good reduction away from primes dividing their level, and have cohomology which can be calculated in terms of automorphic forms, which will usually have bad reduction at primes dividing their level. This makes it easy to verify bad reduction, again using the homological criterion.

A subtlety is ensuring that the varieties have a nonsingular compactification. The $k$-fold fiber product of the universal family of elliptic curves over the moduli space of elliptic curves with full level $N$ structure has a natural compactification as a ramified cover of the Deligne-Mumford moduli space $\overline{\mathcal M}_{1,k+1}$, but this is singular. However, the singularities are toric and should be easy to remove away from the primes dividing $N$. Then as soon as there is a cusp form of weight at most $k+2$ and level dividing $N$, whose level is a multiple of $p$, it will show up in the cohomology (by work of Deligne) and hence this space will have bad reduction at $p$. This is easy to ensure by taking $k$ or $N$ sufficiently large and using formulas for the number of cusp forms.

Because the Hodge structures of Galois representations of modular forms are known, it should be easy to check your motive condition as long as one of the relevant modular forms has weight $>2$ and is non-CM. But because the Galois representation splits into many two-dimensional pieces, your strong irreducibility criterion is essentially never satisfied here.

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