I think fundamentally sims just don't want to/the programming isn't sophisticated enough for them to care for their infants. Adding script mods (I recommend 'slow infant needs decay' by kuttoe and others to limit 'check infant/toddler') does make playing them not as stressful and actually fun for me sometimes, but while I was still vanilla, a couple of things helped:
1. Check that the infant doesn't have any queued or current interactions. If your caregiver is about to feed the infant and they have 'coo with [caregiver]' 'watch [caregiver]' 'what's that?' as an ongoing action, your caregiver won't be able to do anything until it's over. Infant interactions will often override current caregiving interactions too, which contributes to the messiness.
If there are queued/current infant interactions, cancel them all. Enable cheats and use 'reset object' if needs be.
Likewise if there are more than one caregiver in the household, check that the second hasn't autonomously queued interactions with the infant. If this continues to be an issue, set the caregiver you don't need a task, such as upgrading plumbing or going for a run. Queue up several different interactions if needs be.
2. Minimise distractions. I find 'judge decor' to be the biggest culprit. Sims will put down a hungry, stinky wailing infant and go and complain about my farmhouse wall sculpture. So removing any decor likes and dislikes entirely limits this. Also make sure the television or stereo is off. Move them into the household inventory if you have to.
3. Play as the infant and ask to be fed/bathed etc. I'm not a modder and don't have any cold hard data to supplement this, just a hunch really. But it seems like caregiving tasks are less likely to be ignored when you click on the bath as an infant and ask to be bathed as opposed to making a caregiver do it. It also limits the problem of infants having contrary queued interactions as in No 1.
4. Forgot to say but sometimes you have to break down a task into "bite size" pieces. Ie. if I want my caregiver to give an infant a bath, sometimes I have to tell them to pick up the child, then click the "give bath..." option, rather than just clicking on "give bath" only to watch them stare confusedly at the infant for hours on end. Also smaller homes, especially if they lack stairs, definitely makes infant care easier.
Hope this helps!