One thing to bear in mind when doing a self-test is whether your audio equipment (especially if a laptop or phone) is actually capable of reproducing the signals. Just because you press play on a 10kHz tone track it doesn't mean the speaker actually produces the sound.
I have a fairly decent hi-fi kit and even on that a test CD with a frequency sweep shows up "hills and holes" in the output.
Sadly loss of sensitivity to higher frequencies as you get older is a fact of life. Young kids can hear well up into the teens of kHz, old fogies like me do very well to hear 10kHz, and 5kHz is very commonly about the limit. In reality you seldom really "need" much more than a couple of kHz, that's like a flute which has a typical 3 octave range from middle C (256Hz) to C7 (2096Hz)and will let you hear most speech etc well. Tinnitus is another issue to add insult to injury .........
A typical tweeter crossover frequency is around 3.5kHz, and you may well find an older person can't actually hear anything above this anyway, so extra tweeters may or may not be much use. What is probably more use is more mid range, 500-3k or so. According to google, adult male speech is fundamentally in the range 85-155Hz, and female 165-255Hz, generally fairly low comparatively, so anything above 500Hz is not likely to be very much help for those who have trouble hearing speech, although the higher frequencies will make it a bit "brighter" even if the hearing is poor above 1kHz.