Nobody said "time rode horses".
Actually, your answer directly implied it!
When a participle/participial phrase is said to modify a noun, that is tantamount to asserting that that noun is the implied subject of the verbal aspect/force of that participle. ( I use the term 'subject' advisedly: see below)
Thus, for example, in
The boy playing tennis is Sam.
'boy' is at once both the object of modification of the participle 'playing (tennis)' and the implied subject of the action that it denotes (i.e. = 'who IS PLAYING tennis').
You simply cannot have the one without the other!
Thus if you assert, as you did, that a participle (riding) is modifying a noun (time), you must accept that it is also its implied subject, leading to the nonsensical entailment that 'time rides horses'.
As a matter of fact, neither gerunds nor participles have subjects because they are verbals, not verbs.
Many would take issue with that rather sweeping statement!
In the strictest, narrowest application of the term 'subject', it is so. However, I use the term here, as do many grammarians, in its wider sense as the (possibly merely implicit) "performer" of the state/action denoted by the verbal element, so that we may for instance speak of 'you' as being the (implied) subject of the infinitive 'walk' in
I knew that it was hard for you to walk away.
simply as a kind of convenient shorthand for the common sense assertion that it is 'you' (the prepositional object in the surface structure), and not the sentence subject 'I' or some other person, who actually performed the act of walking.
Of course, some may prefer the term 'agent' where a participle or other nonfinite verb form is concerned, but that is clearly an even less satisfactory option, since it can be applied only where the verbal element is active.
The simple fact is that we do not actually possess any completely satisfactory general-use term to express this implicit 'doer' role with regard to nonfinite forms, and are thus reduced to appropriating the closest available analog, to wit 'subject'.
As for your implication that a verb is not a verbal, I think you'll find that you are on equally shaky ground. A (finite) verb is, by definition, a sub-class of verbal, since the term 'verbal' means 'any form functioning like/as a verb' (which, I think you'll grant, finite verbs do!)
Of course, for the sake of referential convenience, you or anybody may elect to make a restrictive use of the term to actively exclude finite verbs (just as some authorities find it convenient to draw a distinction between 'adverb' and 'adverbial'), but it is your choice so to do, not something innately predetermined by the basic meanings of the words themselves!