Nathan, both explanations you're weighing are partly right, and the reason is worth setting out because it comes up constantly.
Organised programmes run to defined upper ages. Bowel screening programmes commonly run to around 74, breast screening invitations commonly stop around 71 to 74, and cervical screening usually ends around 65 with adequate prior negative results. Those upper limits are set nationally, they differ between countries, and above them most programmes will still accept a self-referral rather than refusing people. So your mother's letters stopping is the programme boundary doing exactly what it was designed to do, not a clerk deciding she's had her turn.
The reasoning behind the boundary is the part that gets lost. Screening buys benefit slowly. Finding a slow-growing cancer early only helps if the person has enough years ahead for that cancer to have caused them trouble, and that lag is typically several years for the common screened cancers. The harms, by contrast, arrive immediately: the biopsy, the procedure, the anaesthetic, the hospital stay, and older bodies tolerate all of those less well. Annika has described the mechanism above better than most textbooks manage. So the honest framing is not "old people stop mattering", it's that the benefit line and the harm line cross over at some point, and that crossover depends on the individual's overall health and likely years ahead rather than on their birthday. A vigorous, independent 79 year old may well sit on the benefit side of it. A frail 70 year old with significant other conditions may not.
Which is precisely why this is a shared decision, and why it belongs with the clinician who knows her history, her other conditions and her own preferences. The site's guide to screening in your 60s and beyond works through which checks continue, which taper, and why stopping one is not the same as being written off.
The other half of your question deserves an answer too: plenty does not stop. Blood pressure, medication review, vision and hearing, bone health and falls risk, and adult immunisation all stay on the list and arguably become more valuable with age, because the benefit from those arrives in months rather than a decade. If the mammogram letters have stopped and nothing has replaced them, that gap is the thing to raise at her next appointment.