If things are really warming 20 F degrees in a hour, you're not riding in the cold for very long. I'm not saying tough it out, but am saying you only need something to stay warm for a short time. I recommend a good wind jacket. It's really surprising how much difference that makes. The downside is sweat. I don't care how much fancy pants Gortex this and that it has, it will get wet inside from sweat. But Gortex or similar won't get as wet as something that doesn't breathe at all.

You can't have it both ways. Airflow is what keeps you dry and it's also what cools you down.

I don't have temp increases that fast, but my rides from home either start with a few miles of climbing or about 6 miles of descent. So, I sometimes wear the wind jacket on descents when it's a bit cool. I really like my Castelli Squadra Stretch jacket. It's seems to be mostly rip-stop nylon (doesn't breathe), but has breathable panels on back of the sleeves and down the sides. It packs very small.
https://www.castelli-cycling.com/US/...1511_383_52_XL

It really is surprising how much warmer a wind jacket can be in cool weather. It's my number 1 go-to item for dealing with varying temps.

Another favorite is my Castelli Perfetto RoS 2 Vest. Though I think these have been discontinued. But a vest like this works wonders and packs small. Wind block on the front and vert breathable on the back.

I always wear arm sleeves. Mostly for sun protection in the summer, but heavier ones in cooler weather. I rarely use actual arm warmers, but as many have mentioned, it's an option. Still another is a jacket with removable sleeves.

Something I think is very helpful is a wool mesh base layer. You can get these in a sleeveless style. They add some insulation, but work well at wicking moisture away and hold damp material off your skin. So, if your jerseys gets wet from sweat it won't touch your skin as much and feel as chilling.

In the end, it's several light layers. But I'd try mesh base, short sleeve jersey, arm sleeves, quality vest, and finally a wind jacket. You can stow the vest and wind jacket easily in one jersey pocket. Probably stuff arm warmers in the same one as well. Unzip the short sleeve jersey and ventilate the mesh base as needed when it warms a bit.

Below 60 F my threshold for wearing full tights. And I find I tolerate these even when it's warmer. But not 70 F warm. I mostly base the leg-wear choice on what the temp is for the bulk of the ride. So, perhaps leg warmers is what you would need. But again, for such a short time when it is below 60 F, I think I would just tough it out on my legs. Leg warmers are bulky.

Something to consider is some form of bike storage if you need to shed and stow clothing. I've gotten to where I can manage just using jersey pockets, but in the past I've used various frame bags.

There is no perfect solution. To some degree you'll have to accept being a little cool at the start of a ride and perhaps a bit warm at the end.