I see what you mean, but the actual real impact (business wise) is probably minor / negligible. I don't think we'd have the multi-billion gaming industry we have as it is now if accumulating non-finished games among the majority of the gamer population became a priority (to play and finish them) before and above 'moving to the next thing'. If backlogs of games was something that a lot of people would oblige themselves to finish before moving on to the 'newer' stuff (be it just a game, or a new system) then I'm not sure even the 16-Bit era would have succeeded, or maybe not the 32-Bit era instead. Now of course, I'm pushing it in this example; it's an extreme. The industry would have moved on sure, but perhaps the numbers (on sales of games and maybe systems too) would have been more humble.
But the gist is if we had to complete all the games we wanted to, or at least all the games we actually do own (say, outside of renting; at least during the time when renting was a thing) before allowing ourselves to move on to the next platform / generation (or simply, to newer games on the same platform) then not only would we as gamers have moved on at a much slower pace, but the impact on the industry would have been felt a lot more. I mean gaming companies would have definitely noticed, something like "Why is our sequel not selling?", "The 2nd game was good, why is the 3rd one not selling at all?!", well... because people are busy completing 'x' number of other games before moving to yours, sir. The result would have been companies not taking too much financial risks and producing games (and systems) at a much slower rate. It would have simply been plain and simple not financially viable enough for the industry to move to the incredible rate we've known for decades if the gamers audience wasn't ready and willing to support that rate of change and 'new' every week or month.
I mean I can still remember my younger self back during the NES and SNES years, going to my local games (and movies) store to check out what game I could rent today. The little money I had access to (thanks Mom) was precious and so was my ability to play new games. Did I play the ones (very few) that I owned? Hell yeah, I played the crap out of them. But did I want to try out new stuff? Absolutely. All the time. I would rent a game, sometimes would finish it, sometimes not (was around 50-50 I'd say), but I did not spend my time (and money) on renting the same game over and over until I finished it, otherwise I wouldn't have been able to try out a new(er) game about once or sometimes twice a week (twice a week was pretty rare though, usually just had enough pennies for one rent per week). Now of course times have changed but the principle of not somewhat 'forcing myself' completing a game if I feel like trying something new has remained intact to this day. Maybe it was forged from those early gaming years for me, that's possible. But the reality I believe is that, really, if the majority of gamers were stopping themselves from moving on because they have a backlog then we would really see the impact of that on the industry in general.
How many of us here (just here, on these forums) have maybe 50+, 100+, 200+ or 500+ games on Steam? How many of them were purchased during sales? (Doesn't have to be from sales only but this is just an example), and how many of those games are actually installed? How many of them have never even been installed yet just once? In that list, which ones are the oldest that you haven't bothered trying yet? Ok, but whatever the answers to the questions are (and then some), do you stop yourself from buying a new(er) game that picked your interest or perhaps that is now on sale and you've had your eyes on since the past months or even years? Most likely the answer to this last question is "Oh hell no!". Maybe it is, maybe you do pause and think "Huh actually yeah I'd like to buy Rage 2 but I sort of need to finish the Witcher 2 first, been trying to finish that thing since the past 50 years or so".
Another way to put it for some people (or perhaps a lot) is the way my cousin put it one day (he's not the kind of gamer to 'dwell' on games for long, he plays the ones he likes 'these times', "in the now", then gets his fill of them and moves on): "If I didn't finish it the first time I played it I'll never finish it", and the gist is he'll move on from that one game real quick to the 'next thing' (next thing could be newer or not, "next thing" could simply mean a new game to try, that 'new' game might have been released before the ones he had fun with as well, it doesn't have to be brand new from this week or this month; not necessarily a new game system or a next gen system per se, either). Now myself, I'm not quite like that; but I'm not that far from it either. A bit in the middle, in my case. I can (and sometimes do) come back to a game I have played but haven't finished in my backlog and try to "finally beat it". I have, actually, done that a number of times before. However, I do it mostly for games that I know I was at least a little interested in, in the first place. Not only that but I'm getting older and I have less patience than before, so I am more inclined in trying to finish backlog games mostly if said game can be finished in a relatively short period of time (a few days at most). If it's a really big, lengthy game (usually RPG style) with something like 60, 70 or 80 hours of content then chances are, indeed, that if I didn't get to the end originally when I first played it then I probably won't get anywhere closer to the end on my 2nd, 3rd or 4th attempt (especially considering that "originally", as I say, usually means years ago, and years ago I had more patience and energy; so back then if I didn't do it chances are I won't do it now).
The main point though, is that even if I do check out my games in my backlog (and yes as I said I do that still, to this day) it really doesn't influence my decision on whether or not I want (or "should" / "shouldn't") buy a new game that just came out even if the risk of said new game falling in the backlog list is very much real. But with this said, this is about games. If I go with movies in my case then it's different. When it comes to movies I won't just watch a sequel or a 3rd, 4th or 5th movie in a series just because it's new. If there's a backlog of movies (and usually there's very little of that, since movies are much, MUCH easier to consume than video games since they usually just last 2 hours and you're done with it) then chances are I absolutely will watch those first before watching something that happened to come out last week or last month, or heck, during the year at any point.
So yeah, to give a simpler answer directly to your title? I'd say 'Very Little', on the scale of the industry as a whole, that is. But, maybe, it does influence certain individuals' consumption of games (and the frequency of it) a lot more here and there, sure.