Does the lack of the loops hurt Sakura to the point that theres no point in playing her? I have decent knowledge of the character and will eventually get used to learning how to do it, but what are some ways to play well her without them? What kind of punishes can I use instead of the loops?
Does it hurt? Yes. The loops give her some of that damage/stun that really makes her scary up close. Once they become consistent, your opponents will have to fear consistent 400+ damage for getting hit by a frame trap. And after the loop you get a solid mixup opportunity AND have a chance to stun very soon.
But the damage/stun you get from it is just the reward for playing her correctly, not the actual core of her plan. If anything, I’d say learning how to use your cr.mk -> shouken in footsies is a bigger part of her game. You need to learn how to use your cr.mk effectively, and when to start using her other normals when it isn’t exactly enough (cr.mp, st.lk are good options). Basically, learning how to plan footsies with her basic normals is far and away more important to playing her right.
In the meantime, just doing a simple cr.lk, cr.lp, cr.mk shouken combo is decent enough as a hit-confirm combo. For damage, just stick to “cl.hp xx mp shouken” for a simple combo into 5 frame safe jump, or “cl.hp xx ex tatsu -> ender” for a bit more damage. If you can, try sticking a jump hp or a lv.3 focus in the beginning of those combos for more damage. There are a ton of great Sakura videos that go over some of her combos and intricacies (look up Jay Rego or shano on youtube for good starts).
You can even learn some simpler tatsu loops that are 2 frame links instead of 1 by using st.lp,st.lk or cl.hp. So something like “tatsu -> st.lp, cr.mk, shouken” is a simple 2 frame link that is still relevant. On some characters, tatsu can lead to cl.hp xx shouken, and usually that is most of the damage anyways (any extra iterations only adds a couple more damage in those cases, but it is character specific). And st.lk xx shouken is a very typical ender for tatsu loops anyways, so that one is particularly useful to practice.
Not being able to do loops only slightly hurts your pressure game. Your opponents may be more inclined to go for a crouch tech, or to mash jab, because they don’t worry about the damage as much. Everything else about sakura, however, does not require a tatsu loop, so she is still a decent character to play.
EDIT: I should note: outside of footsies, try learning how to get up close to your opponent and keep pressure up. Breaking people’s defenses down is probably the other half of sakura’s core plan (footsies is just a natural part of everyone’s plan). Again, there are great videos out there for this also, to help you understand it better.