See now in my area teachers don't care for young kids on digital pianos for two reasons. One is all the other buttons and switches that distract young players and the other is lag....
For one, except for perhaps a few poorly thought out off brands, digitals don't lag. Actually, they program the same real world delay that it takes for the hammer to reach the string and have the sound travel 4 feet or so to your ears, just like the real thing. That's maybe a couple of milliseconds, but Casio, Yamaha, Roland, Kawai, Korg & Kurzweil all do that correctly to make it behave as a real piano does.
Second, while crappy toy keyboards have all kinds of deficiencies for learning, most starter digital pianos have a minimum of distractions. For example, Yamaha recently introduced a model P35B. It's the same action as their super popular P85, P95, P105, DGX-640, YDP-141 etc. It's $449 everywhere (and a little less in some places) and it has only a power button and a single function button. Without blinking, I know of at least 6 models that have no more than 7 buttons. If they can't handle 7, how are they ever going to handle 88?? And those distractions...most often that's the metronome.
I know this is the piano forum, but logic should hit you like a baseball bat when you are talking about the moving cost + first tuning cost being more that 30% of the value of the instrument, that it's not "a deal". It's not a good plan.
If your student is a born pianist, you can give them anything to learn on and they'll thrive. 99% of us may learn to like piano but only 1% are born to do it. Give the advantages you can and don't torment the student by asking them learn on the equivalent of a car without first gear and a flat tire. It's like taking expired medicine...it probably won't hurt you but it probably won't help you either. If you don't think it is that bad, I'm here to tell you that it often is.
If you are an experienced pianist, you can easily ride rough over the bumps and play a song, but to a beginner, those are big potholes, loose gravel, and confusing street signs.
If you have a family piano with lots of sentiment, I can understand spending money to keep it going, but at some point and if there is no sentiment, you are not buying someone else's piano, you're buying someone else's problem. Make your first piano a positive thing and take the savings from service and put it towards a better acoustic.