One way they get away with using relatively few tubes is that most of the Boolean logic is done with semiconductor (germanium) diodes, using tubes to amplify the results.
Tubes can also just be very _clever_ in a way that transistors generally are not. For instance, counting tubes that would contain a decimal counter in a single glass bubble (that's quite a few transistors even in TTL, and about twice as many in CMOS). Multiple approaches to building such tubes existed. This is one:
That RAM component is wild. Do you know if there are any equivalent analog RAM components today? By which I mean RAM elements that are similarly capable of storing an analog as opposed to a digital signal.
I don't think I've seen anything sufficiently RAM-like. The classic sense amplifier architecture tends to make this impractical. I guess modern Flash can store a few levels but that's a far cry from actual analog storage.
For a while, you could get things like bucket-brigade CCDs for analog signals (used for reverbs and suchlike). They're still made in small quantities, I think, by boutique operations like Xvive.
Also, there was the ISD ChipCorder, which was actual analog Flash. (Current devices are digital, but the old ones were for a while sold as "MLS ChipCorder" by Nuvoton.)
https://lampes-et-tubes.info/cd/cd053.php?l=e
Or the complicated and impressive storage tubes, that would fit 16 kbits of RAM in a single CRT:
https://lampes-et-tubes.info/sc/sc022.php?l=e
By the way, this website is one of my favorite finds on the Internet and very long-running.