Depends on the game. PS3 of course your forced into it.
But 360 I do depending. Some that I can't as the data isn't there or the laser can't pick it up I just play them off the disk there is no choice but those it can complete and they are noticeably noisy I install them.
Like Baja is loud but I can't install it.
Frontline Fuel of War was more ideal to install then play off the disk.
Others like Forza Motorsport 2 to 4 I went eh I'll install them like I did the additional content disk just to have it all on the system why not.
It varies from game to game for me. If I notice it's noisy I will more so but I still play with music from a phone or something else if not got headphones into an AUX to the TV to hear dialogue/game music if I really care but if I don't care it's music/YT videos from my phone instead.
I use a Xbox 360 Elite with 250GB (hard drive died, used a USB for a bit and used a 120GB ever since, no issues with it other than the disk tray but I leave a disk in and it works fine enough still) about halfway through 360?PS3/Wii gen I think with the HDMI but no wifi so it's the one you'd still need a dongle for. I am happy with it offline anyway as singleplayer games so don't have to care, have ethernet cable hooked it up sometimes for dashboard updates and to prep so can play any game but that's about it.
Also got a MW3 Slim just randomly one day this yeah about what 2-3 months ago now and had a Xbox 360 E since Xbox One.
I primarily use the Elite and now using the E for back compat games or to not have the Elite on the floor/moved into the TV stand among the PS3 Slim/PS4 Base Model/Xbox One X so Elite, PS2 Phat, Wii with GameCube support sort of went anywhere to slot in, sometimes on or in the TV stand, sometimes floor, Switch dock behind the TV.