Great question! I tried to keep it short, but yes of course there are down sides. In fact, mathematically speaking, there literally can't be a perfect voting system. Check out the massive tables in this article for which voting method satisfies which criteria.

The down sides people usually complain about when it comes to approval voting all stem from the same feature, you only get to vote yes or no on any given candidate. If you like both Trump and DeSantis, but it's very important to you that you give more support to Trump, sorry this voting system doesn't have that feature. Similarly, if you don't like Ted Cruz so much that you want literally anyone else to beat him, you can express that opinion by voting for everyone else, but you can't differentiate between all those other candidates.

Every voting system has trade-offs, in this case that troublesome feature (simplicity) is also a bonus. You can't invalidate your approval voting ballot. Any combination of votes is valid. RCV has to either invalidate ballots that don't follow the instructions, or come up with a list of interpretation rules to try and make sense of ballots that don't list the candidates in a neat order. By some estimates the invalid rate for RCV is seven times higher than FPTP. Approval is, again, bullet proof in this regard.

Approval is also extremely easy to understand. RCV seems simple enough, but then it can end up doing very strange things and elect nonsensical winners. The frequency of strange things happening under RCV is debated, but the more competitive the race, the more likely confusing results will follow.

I said I'd keep it short, which is why the first comment didn't have too many details. You can talk election systems for days (notice I didn't talk at all about how these systems translate to proportional methods). In a practical sense, RCV and approval agree on the results the great majority of the time, all the way from winner to loser. In those scenarios, well, why go through all that extra trouble? Keep it simple!