Not true. Free cellular connections exist in some places. And there are non-cellular solutions like LoRA trackers.

That's not true at all! I'm using some 1NCE SIM that cost 10€ for 10 years of connectivity with 500MB of data. For my car I've calculated that it should last about 6/7 years, but even if it would last 1 year, it's still less than 1€/month!

OP "just" needs to find the tracker and selfhost something like Traccar.

I don't know the ins and outs of how they work, and I'm sure there's some catch and they overall skeeve me out a bit, but I have seen a few companies that offer very limited free service, something like 25 mb/month. I don't know how much data a gps tracker would use but that might be doable

I'm sure those companies do everything in their power to get you to pay more than nothing, automatically change your plan if you go over, deceptive emails, etc. so definitely something to be careful about. I also wouldn't have a whole lot of confidence in those companies sticking around for very long.

And while not free, there are some pretty affordable prepaid plans and such that may be competitive or slightly cheaper than what a regular subscription might cost.

Depending on where you live, it may be possible to forego the call plan entirely, in a dense urban area with lots of open public WiFi networks, you may be able to work it entirely off of WiFi.

If you wanted to get real weird with it and jump through the loopholes to get licensed, there might also be some options using ham radio stuff like APRS, though that's probably going to leave your dogs location exposed to any ham who happens to be playing with their radios in your area.

Now I'm not saying that any of that is necessarily a good idea or worth the hassle of setting any of that up, I'm just spitballing some ideas for what someone could potentially do if they did want to homebrew such a thing.