MIDI files as such will be with us for the rest of our lives, and therefore supported by all serious computer systems. (Smartphones are a different type of device in terms of "ideology", not of capacity – the very idea of a "file" is declared old-fashioned there, in order to give the "app" providers more power over the device.) Shortcomings are well-known and a new format for the same purpose has been overdue for decades, but has yet to be agreed upon.

The reason for having an external MIDI chip (on the "sound card") is only about system performance, not per se about quality.

Browsers don't make music. If they receive a file, they will perform the action their user configured them to do, depending on the declared file type: either save them to a permanent file on the hard disk, or to a temporary one and have your OS "open" it immediately after download (as if the file icon were double-clicked). The OS in turn must be told what "open" means, i.e. which software to pass it to. This MIDI player software in turn must be told what "device" to send the individual MIDI commands to - often several are available: built-in hardware, a pertinent USB device, software synth, etc. All these settings will come with defaults chosen by the software at installation time; change them if you disagree.

Joe, please tell us what kind of problems your recipients report.