So, this card has some similarities to Mr. Li, which never saw any play, and this is more expensive. So why do I think The Class Act will be way more popular in criminal decks?
First off, the math is just way more favorable for this new connection. Mr Li has a total cost of 5 (3 credits, click to install and the card itself), while TCA only has a cost of 2 (4 credits + 1 card + 1 click - the 4 cards you draw). This is significantly better, and getting to start your next turn like Andromeda gives you a ton of tempo. Remember, since this triggers at the end of your turn, after hand size is checked, you don't discard the cards.
Furthermore, TCA is way more versatile than Mr Li. The latter only gave you one line of play: spend a click to look at 2 cards, and draw the one you need. The former works with ALL draw engines. Earthrise Hotel, Deuces Wild, Diesel, Build Script, the newly spoiled blue Diesel, all of them will trigger TCA and allow you to draw even more efficiently.
The only disadvantage TCA has is that it only triggers the first time you draw, while Mr Li could be used as often as you'd like. But I consider this a very minor minus.
At 5 influence, this will not see any out-of-faction use, except maybe in Sunny. But expect it out of many criminals!