The current generated in the closed wire loop of course induces a magnetic field $B_{induced}$ opposed to $B$ . The magnetic field would be $B-B_{induced}$.
But by what means you keep the constant or anyother initial magnetic field $B$ ???
By an apparatus that, trying to keep the field constant, increases or decreases the strength of its field from $B$ to $B+B_{induced}$ adding or subtracting energy to or from the wire loop depending on its motion and/or deformation. So the field remains constant :$(B+B_{induced})-B_{induced}=B$. That's the Lenz's Law : saves the energy conservation. This is a mechanism of energy exchange.
By no way the induced magnetic field $B_{induced}$ could be negleted . By principle, a so called induction motor works because of this :
"ELECTRIC MACHINERY" Fitzgerald-Kingsley-Kusko, 3rd Edition, McGraw-Hill Kogakusha.
4-2 INTRODUCTION TO POLYPHASE INDUCTION MACHINES, pages 187,189
"....the induction motor is one in which alternating current is supplied to the stator directly and to the rotor by induction or transformer action from the stator.(page 187)
.... When used as an induction motor, the rotor terminals are short-circuited......The field produced by the rotor currents therefore revolves at the same speed as the stator field, and a starting torque results, tending to turn the rotor in the direction of rotation of the stator-inducing field. (page 189)
Helmholtz Transport Theorem and Faraday's Law
Let $\:\mathbf{F}\left(\mathbf{x},t\right)\:$ be a time-varying vector field defined at points $\:\mathbf{x}=\left( x_{1},x_{2},x_{3}\right)\:$ of a region of $\:\mathbb{R}^{3}\:$ and time $\:t$. If $\:S(t)\:$ is a surface in motion-deformation described by the velocity field $\:\mathbf{w}\left(\mathbf{x},t\right)\:$ of its points, as in Figure, then for the rate of change of the flux of $\:\mathbf{F}\:$ through this surface we have
\begin{equation}
\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{F}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}=\\ \int\limits_{S(t)} \left[\dfrac{\partial \mathbf{F}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \mathbf{w}\right)\mathbf{F} + \left(\mathbf{w}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{F} - \left(\mathbf{F}\boldsymbol{\cdot} \boldsymbol{\nabla}\right)\mathbf{w}\right] \boldsymbol{\cdot} \mathrm d\mathbf{S}
\tag{A-01a}
\end{equation}
or from this equivalently
\begin{equation}
\bbox[#E6E6E6,8px]{\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{F}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}\:=\:\int\limits_{S(t)} \left[\dfrac{\partial \mathbf{F}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \mathbf{F}\right)\mathbf{w} - \boldsymbol{\nabla} \boldsymbol{\times} \left( \mathbf{w} \boldsymbol{\times} \mathbf{F}\right)\right]\boldsymbol{\cdot} \mathrm d\mathbf{S}}
\tag{A-01b}
\end{equation}
Equation (A-01a) or (A-01b) is known as the Helmholtz transport theorem.
['Generalized Vector and Dyadic Analysis', Chen-To Tai, IEEE PRESS, 2nd Edition 1997
equations (6.11),(6.12) page 119.]
Equation (A-01b) is important for the expression and understanding of Faraday's Law in electromagnetics
\begin{equation}
\mathrm{emf} \: = \: - \: \dfrac{\mathrm d\Phi}{\mathrm dt}
\tag{A-02}
\end{equation}
that is, the electromotive force $\:(\mathrm{emf})\:$ along a closed path $\:C \:$ equals to the opposite of the time rate of change of the magnetic flux $\:\Phi \:$ passing through any surface $\:S\:$ whose perimeter is the closed path.
Note that the surface need not be necessarily stationary but can be in motion and/or under deformation.
Now, the magnetic flux $\:\Phi \:$ passing through a surface $\:S\:$ is defined as
\begin{equation}
\Phi\: \equiv \: \int\limits_{S}\mathbf{B}\left(\mathbf{x},t\right)\boldsymbol{\cdot}\mathrm d\mathbf{S}
\tag{A-03}
\end{equation}
where $\:\mathbf{B}\left(\mathbf{x},t\right)\:$ the magnetic-flux density vector.
Replacing $\:\mathbf{F}\:$ by $\:\mathbf{B}\:$ in equation (A-01b) yields
\begin{equation}
\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S}\mathbf{B}\boldsymbol{\cdot} \mathrm d\mathbf{S}=\int\limits_{S} \left[\dfrac{\partial \mathbf{B}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \mathbf{B}\right)\mathbf{w} - \boldsymbol{\nabla} \boldsymbol{\times} \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right] \boldsymbol{\cdot} \mathrm d\mathbf{S}
\tag{A-04}
\end{equation}
and using Maxwell's equations in empty space
\begin{align}
\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{E} & \:=\:-\:\dfrac{\partial \mathbf{B}}{\partial t}
\tag{A-05.1}\\
\nabla \boldsymbol{\cdot}\mathbf{B} & \:=\: 0
\tag{A-05.2}
\end{align}
equation (A-04) yields
\begin{equation}
\int\limits_{S} \boldsymbol{\nabla} \boldsymbol{\times}\left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right] \boldsymbol{\cdot} \mathrm d\mathbf{S} \:=\:-\:\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S}\mathbf{B}\boldsymbol{\cdot} \mathrm d\mathbf{S}\: = \: - \: \dfrac{\mathrm d\Phi}{\mathrm dt}
\tag{A-06}
\end{equation}
By Stoke's Theorem
\begin{equation}
\int\limits_{S} \boldsymbol{\nabla} \boldsymbol{\times}\left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right] \boldsymbol{\cdot} \mathrm d\mathbf{S} \: = \: \oint\limits_{C} \left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right]\boldsymbol{\cdot} \mathrm d\mathbf{L}
\tag{A-07}
\end{equation}
so
\begin{equation}
\oint\limits_{C} \left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right]\boldsymbol{\cdot} \mathrm d\mathbf{L} = \: - \: \dfrac{\mathrm d\Phi}{\mathrm dt}
\tag{A-08}
\end{equation}
and the electromotive force $\:(\mathrm{emf})\:$ along a closed path $\:C \:$ is
\begin{equation}
\bbox[#FFFF88,5px,border:1px solid black]{\mathrm{emf}\;=\: - \: \dfrac{\mathrm d\Phi}{\mathrm dt}=\;\oint\limits_{C} \left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right] \boldsymbol{\cdot} \mathrm d\mathbf{L}}
\tag{A-09}
\end{equation}
Note that the expression $\:\left[\mathbf{E} + \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right)\right] \:$ is the Lorentz force on a unit electric charge moving with velocity $\:\mathbf{w} \:$.
The electromotive force $\:(\mathrm{emf})\:$ along a closed path $\:\:C\:\:$ defined by (A-09) may be separated into two parts
(1) one part called transformer electromotive force $\:(\mathrm{emf})_{transformer}\:$
\begin{equation}
(\mathrm{emf})_{transformer}\;\stackrel{\text{def}}{\equiv}\;\oint\limits_{C} \mathbf{E}\boldsymbol{\cdot} \mathrm d\mathbf{L} =\;-\;\int\limits_{S}\dfrac{\partial \mathbf{B}}{\partial t} \boldsymbol{\cdot} \mathrm d\mathbf{S}
\tag{A-10}
\end{equation}
due to the time rate of change of $\:\mathbf{B}\:$ and
(2) another part called motional electromotive force $\:(\mathrm{emf})_{motional}\:$
\begin{equation}
(\mathrm{emf})_{motional}\;\stackrel{\text{def}}{\equiv}\;\oint\limits_{C}\left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right) \boldsymbol{\cdot} \mathrm d\mathbf{L} = \int\limits_{S} \boldsymbol{\nabla} \boldsymbol{\times} \left( \mathbf{w} \boldsymbol{\times} \mathbf{B}\right) \boldsymbol{\cdot} \mathrm d\mathbf{S}
\tag{A-11}
\end{equation}
due to the motion and/or deformation of the closed path.
This separation of the emf into the two parts, one due to the time
rate of change of $\:\mathbf{B}\:$ and the other to the motion of the closed path, is somewhat arbitrary in that it depends on the relative velocity of the observer and the system (and in any case isn't Lorentz invariant).
'The Feynman Lectures on Physics' - Mainly Electromagnetics & Matter,Volume 2, New Millennium Edition, Basic Books .
Chapter 17 : The Laws of Induction, 17-2 Exceptions to the "flux rule", page 17-2
"We begin by making an important point: The part of the emf that comes
from the $\mathbf{E}$-field does not depend on the existence of a physical wire (as does the $\mathbf{w} \boldsymbol{\times} \mathbf{B}$ part). The $\mathbf{E}$-field can exist in free space, and its line integral around any imaginary line fixed in space is the rate of change of the flux of $\mathbf{B}$ through that line. (Note that this is quite unlike the $\mathbf{E}$-field produced by static charges, for in that case the line integral of $\mathbf{E}$ around a closed loop is always zero.)"
Best Answer
The third of Maxwell's equations (Faraday's law) says that a changing magnetic field has an E-field curling around it. The closed line integral of this electric field is the EMF that drives the induced current in the conducting wire. At a microscopic level, the curling electric field, which has a significant component parallel to the wire, exerts a force on the charges in the conductor.
If your question is "why are Maxwell's equations the way they are?", I'm afraid that isn't a good question for this site.