Classical theory said that this was nonsense since one would never finish the summation (if one were to begin).

It was nonsense in classical theory. Infinite sum has its own separate definition.

I tend to agree and I suppose one could say I see infinity as a verb and not a noun.

There are times in modern mathematics that infinite numbers are used. This is not one of them.

I doubt I'm the best at explaining what limits are, so I won't bother. I may be able to tell you what they aren't. They give results similar to the intuitive idea of infinite numbers, but they don't do it in the most intuitively obvious way. They don't use infinite numbers. They use a certain property that at most one number will have in relation to a sequence. In the case of 1, 1.9, 1.99, ..., this number is two. In the case of 1, 0, 1, 0, ..., there is no such number, so the series is said not to converge.

... "Can we say that 1.999... is a sensible theoretical concept?"

No. The question is "Can we make a sensible theoretical way to interpret the numeral 1.999..., that approximately matches our intuitions?" It wasn't easy, but we managed it.