Of course small keys aren't safe. There are no checks for key nor for the seed words. Only important thing is that it was randomly generated.
If it was, then it would be pointless to check if it some small key due to the huge possible range of numbers it could be. If there was any realistic chance for a key to be small, then random number generator is not doing a good job in the first place and that would be the actual problem to solve.
If it is truly random, then your randomly generated private key could be
1 just as probably as anything else. I prefer to check that my randomly generated key is NOT very small or close to the biggest possible number.
Your address could also be generated by someone else by chance, but this is not considered a problem due to ridiculously small possibility that is still way higher then that your key is 1 or anything else small. There are millions of addresses and there was never a collision and there never will be if mathematics about it hold up. Checking for couple of numbers (even a million is considered a couple of numbers) is unreasonable to say the least. Chances are just so low, that you might as well not waste your time or start worrying for all the more likely things to happen, like getting hit by a lightning a thousand times in a row.
Humans are just not good at understanding such huge numbers so people naturally worry about address collisions and similar stupid things.
Don't worry, your key will not be in the first or the last million keys. It is simply not going to happen if your random number generator is any good.