Rushdown characters have ways to apply pressure (see all those guys with the cool blockstrings like Kung Lao or Liu Kang), Cryo doen's have anything worth a (insert expletive of your choice) that can apply pressure to your opponent. The only thing would be standing 1 for a single hit that can be ducked. Most of his safe moves are "you lose your turn if blocked" kind of moves.
What Cryo has are very good footsies tools (F4 is an amazing tool, D4 into short run gives you plenty of safe stuff to work with) and very good movement (probably the best movement of any non-teleporting character, better than some teleporters, his dash, his slide, the long reaching JIP1...). Most of his moves are designed to advance and reach his opponent, not to win a poke battle or gaining the initiative face to face.
I mean, Kung Lao has the same D4, gets much more off it than Cryo does, runs faster, and his F2 is better than Cryo's F4 in every way. He's way better at footsies than Cryo, but he's still a "rushdown" character, and he has no 50/50 to open his opponent's up with. I think Lao much better fits the label "footsies" IMO
Cryo can now threaten his 50/50 after getting a knockdown much easier than he can anywhere else on screen, at stretch range it's not a 50/50 unless you count slide, and that is a 50/50 where the risk/reward is strongly against you. I think he should always be searching for knockdowns, with Slides, JiKs, whatever, and rushing up and going for the 50/50, or reading armour. He gets much more off his B2 than other variations.
In the neutral you definitely want to abuse your D4 when you are at that range, just aim to get slightly closer is better as it gives you that D3 (which is a 50/50 on hit and -1 on block, although B2 doesn't jail), and this range also gives you F33 which is a nice string to be in range of as Cryo, F122 is nice to be in range of as well and staggering F12 for 0 on block into F33 is dope, but maybe this is kinda gimmicky. B12 is good for chasing backdashes of people trying to get outta that range. D4 into run-up something is definitely great for getting in as you mentioned, and if your opponent adapts to that D4 still guarantees F42 which will open them up into full combo if they tried to do anything like armour through your run-up throw or special, but can cancel into Iceburst for your best pushback string on block.
I've changed my opinion on the nature of Cryo a few times, the recent changes too. This is in my opinion what he does best now. I'm not sure whether or not it foots the bill as rushdown and I'm not going to give it that label because it doesn't quite seem as straight forward as "run up, apply pressure". However he definitely wants to be getting in, whether or not he needs to sit at stretch range with D4 and get in safely or not. Just my perspective, and if I missed anything on Cryo's gameplan let me know
I don't think he was intended to be a rushdown character at any point honestly.
NRS at some point of practically every single one of their HypeStreams, have described him as the Rushdown variation for Sub. That's what they are trying to do with him, they definitely failed at first, not sure if he fits the label even now.