Influence of trend on (supposedly) correlated time series
Q1: What impact does a linear trend have on the correlation between non-spurious time series?
The 4 main measures of correlation are Pearson, Kendall rank, Spearman and Point-biserial (the latter of which is not applicable for this type of problem). For simplicity, I'll only explain how it affects measuring Pearson correlation.
Let's assume $X$ represents a sinusoidal time series without trend: $x_t = sin(t)$, $Y$ represents $X$ with an upwards linear trend: $y_t = t + sin(t)$ and $Z$ represents $X$ with a downwards linear trend: $z_t = -(t + sin(t))$. All series have identical timestamps and have the same unit of measurement (for ease of plotting):

One of the assumption to measure Pearson correlation between two time series is called linearity, that when both series are plotted against one another on a scatter graph, there is a linear relationship:

As you can see, $X$ and $Y$ do not satisfy this condition and so Pearson correlation is the incorrect statistical measure to use, whereas for $Y$ and $Z$ it is. Why though?
Pearson correlation measures the degree to which values deviate from the linear line of best fit between the two series provided. If the relationship is not linear, this relationship will not be measured accurately. This can be shown by plotting the Pearson correlation coefficient as $t$ increases:

Notably, $X$ and $Y$ will also violate the assumption of a monotonic relationship for both Spearman and Kendall rank, and so you cannot measure correlation with any of these methods for $X$ and $Y$ unless transformations of the data are performed to satisfy the underlying assumptions - as you do in the question post.
Linear trends, therefore, don't have strictly positive or negative impacts on measuring correlation. You will just have to react accordingly to the underlying assumptions of the correlation measure you need to use.
Q2: Can I trust the findings of a contractor who doesn't take these type of trends into account?
To paraphrase Hanlon's razor:

It is better to assume ignorance than malicious intent.

If you provide your feedback, the analyst will have an opportunity to discuss through why they chose to pursue a certain route, give them a chance to realise what they did was incorrect, or that they misunderstood the requirements and/or limitations of the project.
Hopefully, this leads to a more positive outcome, given you want the best results and the analyst wants to provide the best service.