Well, I wouldn't say it slows it down "dramatically". That was true back in the USB 2.0 days (max about 25 MB/s). But with USB 3.0 drives, you're usually able to hit a HDD's max speed of 125-180 MB/s.
As for SSDs, the issue is that everyone obsesses over sequential speeds. Yes USB 3.0 will cap the SSD's sequential speeds to about 200-250 MB/s. But unless you're copying large files (e.g. movies), sequential speeds don't really matter. SSDs are slower at reading small files - about 25-50 MB/s, which is nowhere near USB 3.0's speed cap. So in most read/write operations with mixed file sizes, the vast majority of the time is spent reading/writing the small files, so the sequential speeds don't really make much difference.
HDDs are even worse - about 1 MB/s at reading small files (yes, one MB/s). I've got a USB flash drive which hits about 10 MB/s with small files, and I forgot I had it plugged into a laptop I was reformatting and accidentally installed Windows on it. It booted up faster than the internal HDD. So you don't even need a SSD to improve on an internal HDD. If you get a decent USB flash drive (with good 4k small file speeds, not just fast sequential speeds), installed games onto it will be faster than if they're installed on an internal HDD (don't fill up the flash drive though).
This is the USB flash drive I mentioned. Unfortunately Sandisk doesn't make it anymore, and replaced it with a slower version (abysmal small file write speeds).
https://www.amazon.com/SanDisk-Extreme-Flash-Drive-SDCZ80-016G-GAM46/dp/B00KT7DOSE
You can however still get the Pro version, which gets speeds similar to the one I have. But from the reviews it looks like they have reliability issues.
https://www.amazon.com/SanDisk-SDCZ880-128G-G46-Extreme-128GB-Solid/dp/B01MU8TZRV
But a real SSD (with an internal controller) mounted inside the computer will provide the best overall speed. I'm just saying there are a lot of in-between options available which will outperform an internal HDD. And to answer your question, yes an external SSD will be faster than an internal HDD.
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