No Food Mill
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Merely crushing the tomatoes produces a more rustic sauce, because you can't get it as fine, and you don't remove skins and seeds, as straining does.
I'm guessing that the tomatoes are already peeled. So you don't have to worry about anything there. The reason for removing the seeds is that they supposedly turn bitter during cooking. I can't say I've found that to be the case, but then I like a little bitterness to offset the sweetness and acid in tomatoes. So unless I really wanted a smoooooooth sauce, I wouldn't even bother with the sieve and all, just break up the tomatoes before adding them.
Discussions like this always put me in mind of a quote from Mitchell Davis, author of Kitchen Sense. In an interview he said, "I recall another recipe that had you strain a soup twice through a chinois. This sort of excess use of equipment and refinement, I think, is a hallmark of chef recipes. When we eat at home, the soup can be a little lumpy."
One thing that throws home cooks for a loop, particular neophytes, is that that don't realize the differences between a home kitchen and a restaurant kitchen. At home there's a limited amount of equipment and space, and too many cookbooks and recipes are written as if home kitchens had the same kind of lines as restaurants.
That, I think, is Davis' point. And it's one that home cooks should make an effort to understand. There'd be a whole lot less frustration in the kitchen that way.
Coarse Side of a Box GraterDepends on the sauce. If you're making a marinara, or other quick cooker, you want the effective processing you get from a mill.
The best substitute is the coarse side of a box grater -- fast, ubiquitous and inexpensive. A ricer (if you have one) is just a step behind (because it's a slower) than a grater. You could also use a meat grinder with a fairly coarse disc. Your box grater can also substitute for a ricer, but it's not much of a meat grinder.
The second best group of substitutes for a food mill is the blender and food processor. Second best because they're so aggressive. Just stop a little short of a smooth puree. In fact, a food processor is pretty much a mill that goes really fast.
You could also force the tomatoes through a coarse cap chinoise, colander or other sieve. However, unless you keep a very coarse sieve you'll filter out too many solids. The point of milling is that you keep all the solids.
If you're buying canned, there's not much difference to be had from processing whole tomatoes other than a very specific texture which won't hold up to heat, anyway. Simply buy crushed or pureed rather than whole. Problem solved.
If you're making a long-cooked sauce, it's not an issue at all. Just rough chop your peeled, seeded tomatoes and throw them in the pot. They'll dissolve into a smooth sugo after 30 to 60 minutes of simmering and occasional stirring.
This may take a bit more time but I remove the seeds and skins before I begin. I core my tomatoes and place them in boiling water for maybe 15 - 20 seconds. Then I put them directly into a bowl of ice water so they do not cook. I use a pairing knife to peel back the skin of each. When that is done, I cut the tomatoes in half and the move the seeds. You can then just cook the tomatoes and use a blender. Puree for creamy or a nice thick sauce with a chop. Plus whatever ingredients needed. For creamy sauce I use a bit of cream. Chunky, I just use a bit of olive oil. Good luck.