I think that your doubts are the consequence of the historical sloppy introduction of the so-called inexact differentials in thermodynamics.

That is a very poor naming for something which, in general, is not a function of the thermodynamic point and has nothing to do with differentials in any of the possible mathematical definitions. The best way of understanding the status of $đQ$ and $đW$ is by recalling that heat ($q$) and work ($w$) are the integrals over the time of measure of the corresponding fluxes (functions of time). Therefore, in general they are finite quantities, not related to the linear approximation of any function of the state. What the first principle of thermodynamic says is that $q + w$, for all the possible thermodynamic processes between equilibrium states is the difference of a function of state (the internal energy $U$). $$ \Delta U = q + w $$ Of course, there will be pairs of states close enough to allow a good description of such difference through the differential of $U$. But this fact does not transform the corresponding $q$ and $w$ into differentials themselves.

Then, the notation $đQ$ and $đW$ should be interpreted as a fancy way to mean that the corresponding process connects two thermodynamic states such that their difference of internal energy is well approximated by $dU$. But it is only when one substitutes the real non quasi-static and irreversible process with a quasi-static and reversible process joining the same two states that it is possible to represent $dU$ as a differential form.

Therefore, the proper interpretation of something like $$ \oint \frac{đQ}{T} $$ is by looking at this formula as a conventional representation of $$ \int_{t_0}^{t_1} \frac{\dot q(t)}{T(t)}dt $$ where $t_0$ and $t_1$ are the initial and final time of a cyclic process starting and ending at the same thermodinamic state. $\dot q(t)$ is the rate of heat exchanges between system and environment.

A final word of caution is in order about the best mathematical model for the space of the thermodynamic states. Although most of the current introduction to differential forms heavily hinges on calculus on differential manifolds, in the case of thermodynamics, calculus on differential manifolds looks like overkilling. The space of equilibrium thermodynamic states of a simple system described by internal energy, volume and number of moles is in general an open cone, i.e., the positive octant of $\mathbb R^3$, excluding the planes $U=0$, $V=0$ and $n=0$, which a quite trivial structure (one chart is enough to describe it). Calculus on $\mathbb R^3$ is all we need.

Reference: it is not easy to find a description of thermodynamics along this line in introductory textbooks. One noteworthy exception is the Prigogine and Kondepudi book (Kondepudi, D., & Prigogine, I. (2014). Modern thermodynamics: from heat engines to dissipative structures. John Wiley & Sons)

I think that your doubts are the consequence of the historical sloppy introduction of the so-called inexact differentials in thermodynamics.

That is a very poor naming for something which, in general, is not a function of the thermodynamic point and has nothing to do with differentials in any of the possible mathematical definitions. The best way of understanding the status of $đQ$ and $đW$ is by recalling that heat ($q$) and work ($w$) are the integrals over the time of measure of the corresponding fluxes (functions of time). Therefore, in general they are finite quantities, not related to the linear approximation of any function of the state. What the first principle of thermodynamic says is that $q + w$, for all the possible thermodynamic processes between equilibrium states is the difference of a function of state (the internal energy $U$). $$ \Delta U = q + w $$ Of course, there will be pairs of states close enough to allow a good description of such difference through the differential of $U$. But this fact does not transform the corresponding $q$ and $w$ into differentials themselves.

Then, the notation $đQ$ and $đW$ should be interpreted as a fancy way to mean that the corresponding process connects two thermodynamic states such that their difference of internal energy is well approximated by $dU$. But it is only when one substitutes the real non quasi-static and irreversible process with a quasi-static and reversible process joining the same two states that it is possible to represent $dU$ as a differential form.

Therefore, the proper interpretation of something like $$ \oint \frac{đQ}{T} $$ is by looking at this formula as a conventional representation of $$ \int_{t_0}^{t_1} \frac{\dot q(t)}{T(t)}dt $$ where $t_0$ and $t_1$ are the initial and final time of a cyclic process starting and ending at the same thermodinamic state. $\dot q(t)$ is the rate of heat exchanges between system and environment.

A final word of caution is in order about the best mathematical model for the space of the thermodynamic states. Although most of the current introduction to differential forms heavily hinges on calculus on differential manifolds, in the case of thermodynamics, calculus on differential manifolds looks like overkilling. The space of equilibrium thermodynamic states of a simple system described by internal energy, volume and number of moles is in general an open cone, i.e., the positive octant of $\mathbb R^3$, excluding the planes $U=0$, $V=0$ and $n=0$, which has a quite trivial structure (one chart is enough to describe it). Calculus on $\mathbb R^3$ is all we need.

Reference: it is not easy to find a description of thermodynamics along this line in introductory textbooks. One noteworthy exception is the Prigogine and Kondepudi book (Kondepudi, D., & Prigogine, I. (2014). Modern thermodynamics: from heat engines to dissipative structures. John Wiley & Sons)