I am not a physicist. Congenital foolishness has deprived me of mathematical competence. I have struggled to gain some understanding of physics nonetheless (perhaps an effect of the aforesaid foolishness), and have arrived at some level of comfort with the concept of spacetime.
Also, I am disturbed when I see gravity described as a force, as in the first video, showing gravity as one of four forces. Thereafter the narrator pointed out that gravity is just the shape of spacetime in the presence of mass, which means it isn't a force anymore than volume is a force. When I drop a stone into a pond the water rushes away from the stone, the volume of the pond changes shape to accommodate the volume of the stone. This is an energetic event, just as is an apple falling out of a tree and smiting my head, but just as the gravity flinging the apple at me isn't a force, neither is the volume of the stone pushing the water around a force either.
Forces result from mass moving in spacetime, but the geometric shape of spacetime isn't a force in the way electromagnetism, and the weak and strong nuclear forces are, carried by specific particles. If spacetime consists of spacetimeons, then perhaps they are the particles carrying gravity - but that can't be the case, since gravity doesn't travel between massive objects anymore than the shape of the pond water travels to the water and the stone. Where the stone is, water isn't, immediately, and as mass exists in spacetime, spacetime has a shape, immediately. Because of the inverse square law, the effect of mass on a given region of spacetime diminishes the further in distance the mass is from the affected region, but this doesn't happen more slowly across increasing distance. Spacetime is shaped immediately no matter the distance, and the relative mass of the object governs the degree of warping, not adding or decreasing delay.
Particles and forces do take time to traverse spacetime, unlike gravity, which is the shape of the spacetime particles traverse.
I know there is a Planck length. That means that there is also a Planck time, since in my understanding there isn't any distance that can be traversed in the absence of time. I have before attempted to describe how our perceptions of the universe would be different if we perceived space as we do time, and time as we do space. We perceive space as an uninterrupted reach, an entirety from one limit of our perception to the other, but perceive time as a sequence of instants, a series of nows that follow one after the other, with those occurring before this moment and those that will occur after this moment as no longer existing.
This being a somewhat confusing imaginary excursion, and I being particularly incompetent at the moment due to unrelated issues, I won't try to describe that perception here, but will leave it for you to imagine yourself. Perceiving events that occur in spacetime as extending forward and back in time to limits of perception, but space as a sequence of singular points we traverse that thereafter do not exist, and have not come into existence until we reach them, is an interesting exercise (at least, for me), but most interesting about that exercise is that it doesn't alter my understanding of spacetime. Either way I envision perceiving it, it is still one thing, not two separate entities.
So I am quite baffled by your statement:
"At the deepest level of mathematical physics, time does not exist at all."
That seems to me to be the same thing as saying that space does not exist at all, since time and space do not exist at all in my understanding, but spacetime is what exists and we perceiving as we do causes us to consider spacetime as two different things, space and time.
Is that indeed what you mean, or, as I suspect, do you mean something else?
Thanks!