Why do we use plural for indefinite objects?

Building off another question I answered here, I couldn't justify why exactly we say:

I like to ride bicycles.

Instead of:

I like to ride a bicycle.

(This could be anything: "climb mountains", "fly helicopters", etc.)

Both of these forms should refer to some generic, indeterminate bicycle object. But we can't ride more than one at a time, nor do we mean to say that we like to ride them one after the other, so how does the plural form work here?

Even though the second example looks like a valid construction, it just doesn't sound right. I suspect it's because the indefinite article "a" adds something when used in this sense, instead of just meaning to say "a bicycle" in the generic sense, it implies:

I like to ride a (particular kind of bicycle).

However, "I like to ride my bicycle" works. Is it because "my" makes it less generic, or just because it means "the one that is mine"? So why does the indeterminate "a" not work in the same construction?

(The only sense where "a" seems to fit is when using the conditional: I would like to ride a bicycle. But again, this has a slightly different meaning than I would like to ride bicycles, in the general sense.)

How should I understand what's happening in these constructions?

The reason I like to ride a bicycle doesn't work is because the singular indefinite article implies there's just one particular bicycle you like riding.

When you say I like to ride bicycles there's an implied plural indefinite article there, but we discarded that word centuries ago. We're so used to not using it we don't even notice it's not there.

Even though the article is missing, we understand as if it were there; we don't suppose OP likes riding more than one bicycle at a time - he likes riding [some, most, many, all, any] bicycles.

I think the success of this usage encourages us to discard the singular definite article in various related contexts. So I like to play the piano, is increasingly giving way to I like to play piano (even more noticeably with the more modern I like to play guitar).

It seems to me that your first example ("I like to ride bicycles") is using a workaround for the missing plural indefinite article in English. Effectively there is an implied plural indefinite article there. So both sentences are, in effect, using the indefinite article. That being the case, why singular vs. plural?

That would be because the indefinite article can indicate different things about the noun. Specifically,

"It may be something that the speaker is mentioning for the first time, or its precise identity may be irrelevant or hypothetical, or the speaker may be making a general statement about any such thing."

It would seem that the plural is preferred in the latter case, when making a general statement. You can, though, make a general statement out of the singular by adding a qualifier or using a conditional:

I like to ride a bicycle when I am at the beach.

I would like to ride a bicycle at the beach.

That seems to be the distinction: unadorned singular indefinite articles don't work as a way to generalize.

I am not a trained linguist, so I will likely need some correction on this, especially the conjecture on the implied plural indefinite article and the lazy generalizations about singular vs. plural.

‘I like to ride bicycles’, while being grammatical, strikes me as odd. It suggests that I like to ride lots of different bicyles. If I want to say that cycling appeals to me in general, then ‘I like to ride a bicycle’ seems to meet the case, although it might more often be modified as ‘I like to ride a bicycle from time to time’ or ‘I like to ride a bicycle when it’s not too cold.’ The indefinite article can be used to mean ‘any example of something’ (‘How English Works’, Swan and Walter) and that’s exactly what it’s doing here. After all, we’d say ‘I like to carry an umbrella when rain is forecast’ and not ‘I like to carry umbrellas when rain is forecast.’