The next question is whether there are programs with lower time bounds that can be made arbitrarily large. It's not precisely the same as "arbitrarily large",

The next question is whether there are programs with lower time bounds that can be made arbitrarily large. It's not precisely the same as "arbitrarily large", but this is the practical result given the busy beaver function. Following from that, yedidia and Aaronson proved the existence of BBs outside of ZFC that are smaller than almost all real programs. I'd consider that essentially what you're complaining is unproven here.