1. That's exactly what General Relativity does: It describes spacetime geometry in terms of the metric tensor, that is a solution to Einstein's equations.

2. Einstein's equations show exactly where spacetime ``comes from''.

3. They also describe spacetime singularities as inevitable consequences of the property of the Einstein-Hilbert action of describing a gauge theory with noncompact gauge group, the group of diffeomorphisms.

Really the thread question “Limitations of General Relativity in Explaining Time” is rather vague, in the GR - and in every not too strange physical theory, including the GR, – besides “time” also “space”, both of which compose Matter’s spacetime, are fundamentally involved, and so in the question both phenomena/notions should be.

At that both -“ Space” and “Time”, are really fundamental phenomena/notions, and, as all other such ones, first of all in this case, “Matter”– and so everything in Matter, i.e. “particles”, “fundamental Nature forces” – and so “fields”, etc., “Consciousness”, “Energy”, “Information”, are fundamentally completely transcendent/uncertain/irrational in the mainstream philosophy and sciences, including physics, - while really they all can be scientifically defined only together;

- and so in every case when the mainstream addresses to any really fundamental problem, the result is completely inevitably is transcendent/mystic something.

Including that are the GR postulates that interactions of bodies in gravitationally couopled systems are intercations in systems “mass-spacetime-mass”, when not only time, but also space is curved; and this curved spacetime forces the bodies to move along curved geodesics.

The fundamental phenomena/notions above can be, and are, really scientifically defined only in the Shevchenko-Tokarevsky’s really philosophical 2007 “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

Compact, but essentially clear and rather in detail introduction in the conception see in

https://www.researchgate.net/publication/387933457_A_response_to_question_What_is_the_The_Information_as_Absolute_conception;

- and more concretely at the definitions application in theories of what exists and happens in Matter, i.e. in physics, in the Shevchenko-Tokarevsky’s Planck scale informational physical model, which is based on the conception, two main papers are https://www.researchgate.net/publication/391209088_The_Informational_Conception_and_Basic_Physics, [ in the paper secton 2.9 “Mediation of the forces in complex systems” can be passed since this is more comprehensively given in section 6.“Mediation of the fundamental forces in complex systems” in other paper below] , and

https://www.researchgate.net/publication/367397025_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics

Though these GR postulates weren’t completely new in 1916, in 1908 Minkowski in what now is called “SRT” also postulated some interactions of material objects and space/time/spacetime - that in moving purely material inertial reference frames “time is dilated” and “space is contracted”, etc., what also really fundamentally cannot happen in Matter.

More see the links above, to read SS posts in

https://www.researchgate.net/post/Does_time_flow_more_slowly_in_higher_energy_density_environments/2 ,

https://www.researchgate.net/post/What_to_think_about_loop_quantum_gravity/2,

https://www.researchgate.net/post/Is_this_research_paper_solving_gravity_experimently

- it is useful as well.

Cheers

Stam Nicolis So, time is just what clocks measure. Is there no philosophical concept of time?

No. There's no such notion as ``a philosophical concept of time" (or space). Or if there is, it's the outcome of trying to express in words what the mathematical description could mean. Just like any other concept.

Stam Nicolis If Einstein's clock-time is all that exists, why does our experience of time bend not with gravity, but with love, boredom, fear, and hope?

This

“…So, time is just what clocks measure. Is there no philosophical concept of time?….”

- is indeed quite natural question relating to the standard physically “time is just what clocks measure” - in this case it is quite evident that some set of some pieces of metals and non-metals “clocks” by no means are “times”,

- and this

“…why does our experience of time bend not with gravity, but with love, boredom, fear, and hope? ……”

- looks as something like to the first quote - love, boredom, fear, and hope also quite evidently aren’t “times”.

Including both quotes above - and the “our experience of time bend… with gravity” – are like to the above in that all [clocks, bending gravity, love, boredom, fear, and hope] fundamentally are nothing else than some fundamentally subjective humans illusions, whereas for Matter it is all the same what human do, including at thinking.

An that yeah, “Time/time” is a really [absolutely] fundamental phenomenon, so is principally subject for study by the science “philosophy”, as that fathers of philosophy established more 2000 years ago,

- and just first of all philosophy must scientifically defined, in this case “Time”, so that this definition would be base for other sciences, including, say, if that would be so really illusory and fundamentally wrong “the time is just what clocks measure” would not be in physics.

But till now the mainstream philosophy by no means is a real science, and “Time/time” – and all other really fundamental phenomena/ notions - are completely transcendent/irrational in the mainstream.

Philosophy can be a real science only basing on the SS&VT “The Information as Absolute” conception, the links see in the SS post above, where most of fundamental things are rigorously scientifically defined/clarified.

Cheers