This much standing water tells me that you need perforated drainage tile* or pipe* in addition to your drain(s).
This will pull water from the ground after it's soaked in as well as removing surface water. Simply having a storage tank for water doesn't look like it'll make enough of a difference for you. The drain you have is designed for quick removal of water, which you definitely need, but adding the perforated pipe will help remove the water that doesn't directly lead to the drain.
You will need to put tile under all the lawn that is flooded, so you might have to tear up more of the grass than just replacing the current system.
A well designed garden landscape can transform your outdoor space, but it is important to protect your garden from changing weather conditions. Using perforated pipes discreetly in your garden landscape can help to absorb excessive rainwater, draining it away from your garden and property. By installing perforated pipes under the ground, excess water can be drained away so that your garden landscape is not at risk of damage.
Replacing each of the 90 degree corners with 2x 45 degree elbows will help the flow and let you remove blockages easier, but it's not the whole problem with your flooding. You just don't have enough points of entry to the drainage system, which is the real problem. Adding more drain grates won't hep much, since you also need to deal with the water soaking into the ground. And adding more grates won't help much if your current pipe is already at full capacity which is likely why the grate is submerged.
(Looking at your 90 elbows again, you can probably only replace 1 with the a single 45 and turn the drain accordingly. Or if it's actually a T, replace it with a Y*.)
Simply putting in larger drain grate won't solve the problem of not having a pipe capable of handling the extra flow. Since your current drainage can't keep up with the rain, causing the drain to be submerged, it sounds like you need to increase the diameter of your pipe anyway. You mention that even a moderate amount of rain causes the drain to be submerged, so I'd suggesting looking to increase the pipe by at least 1.5-2x what you already have.
And, for a complete answer, I'd definitely suggest trying to unclog your current system first, like FreeMan says. This kind of hinges on whether the drainage system ever worked well to begin with. If it could never handle this amount of water, you will need to increase the size of your pipe. And since you have standing water after the drain catches up, you still need need perforated drainage tile/pipe to let your lawn dry out.
*These are examples of what to look for, not a store or brand recommendation.