Yeah, you will pretty much end up crushing the blacks, or having to at least do a separate pass for the opening space scenes/crawl.

Also, yeah it is really important to watch the scenes on an actual television, or a monitor that can really do Rec.709 as it will look totally different to a computer monitor. A lot of monitors are set to a gamma of around 1.8 vs around 2.2 on most TVs.
Also if driving a TV from your PC, it is easy to get PC HDMI levels screwed up vs TV HDMI levels, plus different players will give completely different colour/gamma results, be sure to use one that respects the source data correctly. You will find with some files that WMP, VLC, Quicktime Player etc. will give different results, and that DVD or BD playback software may require a different calibration routine as some of them bypass the ICC calibration.
It is a bit of a minefield on the PC, it can be done, but you have to really check that it is outputting what is expected from the software of your choice.

For playback from PC, I use a dedicated broadcast output card, or external box, they aren’t expensive and once calibrated, they stay that way.

An i1 probe can be had pretty cheap too, and then you can calibrate your TVs and your monitors and get consistent results.
If anyone finds the probes to be expensive, it can effectively cost you nothing, calibrate everything you own, then resell the probe, they hold their value pretty well, and you end up with calibrated gear for almost free.
Sure your gear will drift over time, but you will at least start off with a calibrated system.