I'm disagreeing with the general statement that heavier bullets are designed for big animals like elk, not small animals like deer.
Projectile engineering is much more technical than that.
Many bullets available heavy-for-caliber (example - like Nosler Ballistic Tip Hunting) are designed with thinner jackets at the nose to expand more quickly on lighter game. In fact, many shooters complain that Nosler Ballistic Tip open up too easily, even though they are available in rather heavy weight-for-caliber. Opening up quickly isn't just weight, it is also jacket construction.
Conversely, many bullets (like Nosler Accubond) are designed with thicker jackets to expand more slowly to get deeper penetration before opening up, if ever opening up. Even if an Accubond is lighter weight, it will open up more slowly than a Ballistic Tip Hunting.
My choice of BT and Accubond as comparative examples will help you understand that jacket construction is extremely important, in fact vitally important. Even if you don't use Nosler, comparing the construction of those two bullets (and then seeing that BOTH are available in the full range of bullet weight) are good for personal education. It's not just the weight, it's also the jacket.
Oh, but it's not just jacket design. Some hunting bullets are bonded, some not. And some use the cannelure to help lock the jacket to the core, even if you don't use the cannelure to crimp the case mouth. (Hornady Interlock is an excellent example, in Hornady's words the cannelure "works with InterLock® ring to ensure the core and jacket remain locked during expansion".)
So how does bullet weight make a difference, even if construction is the same? Heavier bullets often carry more energy while travelling slower. If that's what you need, bullet weight may be the answer.
So it's not just bullet weight. It's not just jacket. It's not just core. You need to learn everything about the design of each bullet when making your choice.
I know I disagree with some of you, but no animosity intended. Any of you who think I'm just a stupid idiot who doesn't know that heavy bullets are always designed for elk, you may ignore me. I won't be hurt. I like all gun guys and especially those who hunt. Even if we disagree, we're still brothers here.