Hi Vowel Lady
Your neighbor friend is lucky to have you and your husband caring for him so much - especially if he's on his own. Sounds like your art of gentle persuasion is gradually having an effect. You're right to start slowly with educating him about the need for supplementation- also if he introduces only one new thing at a time it is easier to monitor any effect - good or bad.
It's important that he have any testing done before he starts the multi-vitamin as this will give a baseline for his status. Also the multi-vitamin can affect the test results. If he has already started taking them then he should stop them for about 5 days so they don't affect the test results.
If your neighbor is open to watching just one video I would recommend this one. It's about Vit B12 - but that is one of the more essential vitamins for seniors - particularly if they are prone to falls. You're probably familiar with MrsD's threads on B12 so I won't go into details here.
https://www.youtube.com/watch?v=BvEizypoyO0
Maybe watch it yourself first and pick out a few essential parts to show him to start with (you can use the 'time lapse' bar at the bottom of the video to stop/start where you want). It may help to convince him enough to watch the whole thing.
If you do manage to convince him to ask his doctor for testing and the doctor does do it then also suggest he get a copy of the results for himself.
Another useful site if your neighbor is skeptical about vitamins and thinks it's a waste of his money or he doesn't know what to buy is Labdoor which has done independent testing of some supplements. Here's the link:
https://labdoor.com/rankings/multivitamins
Hope this has been some use to you for your ongoing efforts to help your neighbor stay well.
Added: Vowel Lady - just reread your post and noted he is going for a full neuro exam in a few weeks. If he's planning on going alone it may be useful to gently suggest he have a support person with him - if for nothing else to help remember what was tested and anything the neuro's may say.