Estimating $TFP$ using Cobb-Douglas production function Suppose we want to estimate total factor productivity (TFP) under time series framework. Let assume that the production function is given in the Cobb-Douglas form, i.e.
$$Y_t=A_tK_t^\alpha L_t^\beta,$$
where $A_t$ is the total factor productivity (TFP), $K_t$ is capital stock and $L_t$ is labor. After log-linearization the empirical model is given (time series are $I(1)$, therefore we take in the first-difference)
$$\Delta ln Y_t=\mu+\alpha \Delta lnK_t+\beta \Delta  ln L_t+\epsilon_t,$$
where $TFP_t=\widehat \mu+\widehat \epsilon_t$;$\widehat \mu$ is average factor productivity and $\widehat \epsilon$ is the deviation from average over time.
Questions

In the above specification, by construction, $\mathbb{E}\widehat \epsilon_t=0$ (moment condition). It means that by estimating the above specification yields to $TFP_t=\widehat \mu+\widehat \epsilon_t$, which hasn't any trend (upward- or dawnward-sloping). However, it is natural to assume that $TFP_t$ time series can have a time trend. If no, what is the economic interpretation of no-trend in the $TFP_t$? Is it possible to derive time-trend in the $TFP_t$?
What is the interpretation of the estimated $TFP_t$?
Suppose the estimated $TFP_{2020}=2$ or $TFP_{2019}=3$, how these numbers can be interpreted?

Thanks in advance!
If you want to control for some deterministic trend you could add a trend term to your equation. For example, if you think there is a linear trend you can add $\gamma t$:
$$\Delta \ln Y_t= \mu+ \gamma t + \alpha \Delta \ln K_t+ \beta \Delta \ln L_t+\epsilon_t.$$
However, this being said since the equation is already estimated in first differences you have to assume there is a trend in the growth rate of TFP not just in TFP.
Regarding the interpretation, the TFP itself is unitless and there is no agreed upon measurement but higher is better. What studies usually do is not to look at TFP itself but at its growth. So in this case if TFP in 2020 is 2 and TFP in 2019 is 3 the growth rate would be $\frac{2-3}{3}=\approx -0.3$. That would mean that economy technologically regressed somewhat between the two years.
This being said one has to be very careful when interpreting TFP as its estimates can also partially reflect changes in returns to scale, markups due to imperfect competition, or gains from sectoral reallocations (see this World Bank brief). This is especially problem when you look at an aggregate data rather than at firm level panel data.