The biggest issue with the CR-V is the 1.5T engine which is prone to oil dilution (though Honda claims to have fixed it). CVT issues were mostly resolved back in 2015 but you are correct to be weary. For reliability, your best bet would be to get a used RAV4 with the 6 speed automatic (around 2016 2017). 

Keep in mind the 8 speed automatic was released in the RAV4 for the 2019 model year, and the CVT in the CR-V for 2015. Toyota generally does have transmission problems the first year they are released in fact even Toyota Honda have unreliable models in the first year of the generation but tend to fix issues after 1-2 years. I personally would get a used RAV4 but you can decide with the information above. 

If you trust owner complaints, transmission issue complaints stopped rolling in around 2015 in the CR-V, and in the RAV4, there is an extremly high number of transmission hesitationcomplaints for the 2019 model year, but none for 2020 (source: carcomplaints.com) . As you see, they fixed MAJOR short term issues in the first year. Long term reliability is not known as its a 2021 model, but my bet would be toward the traditional auto in the RAV4. 

Between those two, I’d go with the RAV4.  2021 MY is the 3rd year of that generation and the RAV4 is Toyota’s best seller so any first year issues (2019) are usually resolved by now.  For longevity, I’d still be weary of the CVT in the Honda which would lean me to the RAV4.

For the base model, I would go CRV. Cheaper, quieter, and 'punchier'. Unless you plan on towing, I would say go CRV. The only thing I dislike about the CRV is the turbo 1.5, but I haven't heard of any problems yet.

(But if you are towing, don't even look at these crossovers, go look at compact pickups)

My take: … for me, no turbos, no CVTs, no direct fuel injection only, … so duals are only choice these days. That means RAV4. In fact, … a family member splurged on one, ‘19 model. No problem at all - so far after 1-1/2 years, other than more I look at it, … more I get scare when something does eventually go off! Have-a-lookie at this - just an engine for now:

https://toyota-club.net/files/faq/18-03-20_faq_df_r4_eng.htm

F.S.

To my knowledge, both Honda CR-V and Civic share the same i-VTEC turbo engine.

This engine is known to have gas leaks into the oil system.

So probably RAV4 is a safer choice.

I've driven both the RAV4 and CRV regularly for the past 5 years.  Specifically the 2015 models.  

The RAV4 feels sportier and more fun, probably because of the transmission. The RAV4 has about 50K.

The CRV feels more "luxurious" for lack of a better term.  The CRV has about 90K, and the CVT is still strong.  It's weird, but has worked well.  My first time using a CVT.  TMK, the quality of the CVT seems to be up to par.  IMHO, it's whether or not you like how it shifts.  In theory, CVT's seem simpler to me.

IMHO, I don't think you can go wrong with either one (assuming you get naturally aspirated), even with the CVT.  I recommend test driving both to get a feel for both.  Even though they are in the same category, they drive significantly differently. See which one you like the feel for better.

I personally like the CRV for longer free road trips.  And RAV4 for curvy and winding roads.