Conceptual Question about Independent/Matched Sample t Test Calculations

I'm trying to better understand something conceptually, and was hoping if a few users might be able to explain it I could understand it better.

For the calculations of a test statistics for an independent sample t test, the mean of sample 2 is subtracted from the mean of sample 1 (then 0 is subtracted to represent the mean differences determined by the null hypothesis). This creates a difference measurement in the numerator.

For the denominator, the variances of both sample 1 and sample 2 are pooled/added. So two measurements of variances/stdevs put together is the denominator. Where I am starting to hit a mental block is how this relates to a matched sample t test, where there is only one measurement of variance/stdevs in the denominator.

Both t tests both measure differences, but in one case, there are two forms of variance in the denominator, and the other there is one.

I understand that we need to take into account the variance from both sample 1 and sample 2 in the independent sample test, whereas for the dependent it is a matched design so the variance of the difference will suffice, but why keep both variances for an independent sample t test in the denominator, as opposed to some composite variance? How does it work, that two tests that are so extremely similar, one relies on a denominator that has the potential to be much larger than the denominator of another test?

I'd love to hear any explanations you all have!

Thank you everyone!