There are, in fact, a lot of 32-bit ARM chips still being deployed today. Yes, arm64 is usable, but using e.g. a Beagleboard- or even Raspberry Pi-class device still often makes sense (for cost or compatibility reasons.)
No 2016 beagleboard implementation will be running BigCo's finances and be irreplaceable in 2038. Lets be realistic here.
Those 70s and 80s programmers were working on mainframes with multi-decade depreciation. We work on servers and projects with 3-5 year deprecation when we aren't working on evergreen cloud configurations. Not to mention we've already standardized on 64-bit systems, outside of mobile, which is soon following and has typically a 2 year depreciation anyway.
But some industrial- or military-spec ARMv7 core running a critical embedded system or two, in 2038? Twenty-year design lifespans (often with servicing and minor design updates) are definitely not unheard of, and successful systems often outlive their design lifespan.