It depends! If your entire loading time of the page within the process stays within a second, I would recommend to implement similar controls of the web page. Your image of Firefox implementing the range input is really bad, and if it can be avoided it should.
Jacob Nielsen, a usability guru, wrote an article of Response time limits on applications and web applications saying:
0.1 second: Limit for users feeling that they are directly manipulating objects in the UI. For example, this is the limit from the time the user selects a column in a table until that column should highlight or otherwise give feedback that it's selected. Ideally, this would also be the response time for sorting the column - if so, users would feel that they are sorting the table.
1 second: Limit for users feeling that they are freely navigating the command space without having to unduly wait for the computer. A delay of 0.2-1.0 seconds does mean that users notice the delay and thus feel the computer is "working" on the command, as opposed to having the command be a direct effect of the users' actions. Example: If sorting a table according to the selected column can't be done in 0.1 seconds, it certainly has to be done in 1 second, or users will feel that the UI is sluggish and will lose the sense of "flow" in performing their task. For delays of more than 1 second, indicate to the user that the computer is working on the problem, for example by changing the shape of the cursor.
10 seconds: Limit for users keeping their attention on the task. Anything slower than 10 seconds needs a percent-done indicator as well as a clearly signposted way for the user to interrupt the operation. Assume that users will need to reorient themselves when they return to the UI after a delay of more than 10 seconds. Delays of longer than 10 seconds are only acceptable during natural breaks in the user's work, for example when switching tasks.