As above, the ultimate solution is a full-on "Smart Bike" like the Wahoo Kickr Bike, Tacx Neo Bike, Stages SB20, Wattbike Atom. But unfortunately those are well above your intended budget. Cheaper stationary bikes tend to be more like gym "spin bikes" with manual resistance controls etc. They are not really much like riding a road bike IME.
Given your budget, your best bet is to get a direct drive smart trainer (e.g. Wahoo Kickr Core), which will actually fit a 7-speed cassette with the appropriate spacer. So you should be able to set up your old bike on that. A wheel-on smart trainer would be your cheapest option (e.g. Wahoo Kickr Snap) but they make a lot more noise and are less accurate in power. But I would still go that route over a cheap stationary bike.
Yet another option to consider is the Peloton stationary bike, which is kind of halfway between a gym spin bike and smart trainer. But also pretty expensive and not compatible with 3rd party Apps like Zwift.