Not specifically to this scenario, but I have to disagree with what has been said by a few people here. I do not think I would necessarily call medical control for this patient unless something in her presentation made me uncomfortable leaving her alone. Crochity, you say that anyone over 70 with a medical complaint should be transported. What are you defining as a medical complaint? I don't know to many people over 70 who don't have a medical condition that they live with day in and day out, that sometimes flares up.
Here is what I would do/ want to know were I in a similar situation
First, instructor, you say carotid pulses are present. What about radial pulses? I assume they were as later you state a pressure with 108 systolic. If I am having trouble ascultating a pressure, I attempt to get one by palpation, then either have my partner, or the monitor try. After obtaining baseline vitals with the pt on the floor/ how we found her, and doing a basic assessment, I would then assist her to sitting. At this time I would reassess vitals, check how she is feeling, and do a more detailed exam. If nothing is hurting and vital stay the same, I would then assist her to standing. How does she normally walk? Independently or with a walker? Does she use a wheelchair for distances? If she is is able to ambulate around her house (get to and from bathroom/ kitchen), I would be comfortable letting her refuse, while encouraging her to call her doctor for an evaluation for frequent falls, and having a plan for friends/ family to come over to stay with her for a while. (NOTE, this is based upon her earlier statement that falls are due to feeling unsteady on her feet/ increased weakness, NOT dizziness...). If she does not have a walker, she probably needs an evaluation for one, which they will not do in the ER.
If I get pt to standing and she is unable to ambulate independently/ with her walker, I would have her sit back down and we would have a discussion. I would tell her I don't feel safe leaving her at home alone when she can't get around. What if she needs to use the bathroom again, how will she get there? I will try and point out realistic problems that she can/ will have if she is unable to walk. I will encourage her to call her family/ friend, and if it is OK with her, explain to them what is going on, and try and have them convince her to be transported. If pt is adamant she will not go by ambulance, I will bring up the option of having family/ friends drive her to the ER. If she is still refusing transport, I would document and have her sign a refusal, and have someone else present sign as a witness. In this scenario, I would consider calling med control, but unless she has a physical complaint other than increased weakness, I doubt they are going to want anything to do with it.
As far as how to teach your student/ new person this, sometimes the best thing to do in these situations is watch. I am still a relatively new provider, and I am thankful that I had the opportunity to watch different EMTs and medics with much more experience then I deal with these types of scenarios, so I got to pick out parts that each one of them used, and put them together in my own style. Then when I became more comfortable, I started running the entire call on my own, with them watching (actually they said "we will not do anything including getting vitals unless you specifically ask us to") only jumping in if I was going to miss something major.