Keeping in mind the various points of view expressed in the OP and the comments above, it would be best if we approach this from the point of view of etymology.
The word "sarasvatī सरस्वती" etymologically means one who has a lot of speed and flow, from the root dhātu "sṛ सृ".
So the word would be originally used to describe a mighty river with a lot of water and volume. This has been confirmed from numerous hydrological studies and satellite imaging studies that show a huge dry riverbed that spans up to 12 miles in width at some places.
Michel Danino, "The Lost River: On the Trails of the Sarasvati" (Book)
The evidence from the Rig Veda also shows that the rishis knew Sarasvati as a physical river. Clearly, they attributed metaphysical characteristics to the river because of their spiritual philosophy and worldview. For example, even in the oldest part of Rig Veda (RV 6.61.2) she is described as a river with a very strong waves:
इयं शुष्मेभिर्बिसखाइवारुजत्सानु गिरीणां तविषीभिरूर्मिभिः ।
पारावतघ्नीमवसे सुवृक्तिभिः सरस्वतीमा विवासेम धीतिभिः ॥
iyaṃ śuṣmebhirbisakhāivārujatsānu girīṇāṃ taviṣībhirūrmibhiḥ ।
pārāvataghnīmavase suvṛktibhiḥ sarasvatīmā vivāsema dhītibhiḥ ॥
"She breaks up the rocks with her immensely big and powerful waves just as a digging tool digs up the soil. We worship Sarasvatī, who destroys things on her banks with high speed, for protection with our words and actions."
But then, the same hymn talks about the goddess (RV 6.61.4):
प्र णो देवी सरस्वती वाजेभिर्वाजिनीवती । धीनामवित्र्यवतु ॥
pra ṇo devī sarasvatī vājebhirvājinīvatī । dhīnāmavitryavatu ॥
"May the Goddess Sarasvatī, bounteous with food and riches, the protector of rishis, protect us."
The same goes for other rivers mentioned in the Vedas. They are simultaneously rivers and goddesses. However, their names originate from them being rivers first.
Also, in regards to this statement in the OP:
We have not maintained a single religious place as a memorial of that great river of which we claim to be descendent.
I don't think that is true. All along the course of the river, when the immense volume of water and the meandering caused oxbow lakes to form, at every such lake, there was a holy town. Such towns are recognizable even today in their names containing "-sar" and "-pat".
EDIT: I just remembered a very interesting and unique ritual called the "सारस्वत सत्त्रम् sārasvata sattram", a type of dīrghasattram (i.e. a ritual that goes on for very long periods) which is dedicated to the river goddess Sarasvatī. It is thought to be instituted in the later Vedic period when the river no longer reached the sea, but instead stopped at a place called Vinaśanam. In this ritual, the entire yajñaśālā is built on wheels and is intended to move along the southern bank of the river from the Vinaśanam all the way to the northernmost point of its course, Plakṣa Prāsravaṇam. It would appear like a gigantic rathayātrā, and it is estimated to take more than 20 years to complete.
This is another evidence for actual existence of the river.